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November 30, 2023
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We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p -Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.

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November 28, 2023
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On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p \mathcal{D}^{{1,p}}_{0} into L q L^{q} and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 𝑝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 𝑝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 𝑝-Laplacian with sub-homogeneous right-hand side, as the exponent 𝑝 diverges to ∞. The case of first eigenfunctions of the 𝑝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.

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Open Access
November 23, 2023
### Abstract

We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of the recent work [J. A. Carrillo and R. Shu, Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D, Comm. Pure Appl. Math. (2023), 10.1002/cpa.22162] in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that, for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two-dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one-dimensional vertical collapse is not allowed. Collapse to lower-dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.

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November 20, 2023
### Abstract

We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.

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Open Access
October 27, 2023
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In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p -Laplacian-type diffusion operator with 1 < p < ∞ {1<p<\infty} . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi’s truncation and freezing coefficient methods.

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October 27, 2023
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In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler–Lagrange equation whose graphs produce a foliation. Then the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory.

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October 27, 2023
### Abstract

Optimization problems on probability measures in ℝ d {\mathbb{R}^{d}} are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c ( x , y ) = ℓ ( | x - y | ) {c(x,y)=\ell(\lvert x-y\rvert)} where ℓ : ℝ + → [ 0 , ∞ ] {\ell:\mathbb{R}_{+}\to[0,\infty]} is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non-existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper, we characterize the relaxed functional generalizing the results of [4] and present a duality method which allows to compute the Γ-limit as N → ∞ {N\to\infty} under very general assumptions on the cost ℓ ( r ) {\ell(r)} . We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass < 1 {<1} . In a last part, we study the case of a small range interaction ℓ N ( r ) = ℓ ( r / ε ) {\ell_{N}(r)=\ell(r/\varepsilon)} ( ε ≪ 1 {\varepsilon\ll 1} ) and we show how the duality approach can also be used to determine the limit energy as ε → 0 {\varepsilon\to 0} of a very large number N ε {N_{\varepsilon}} of particles.

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October 27, 2023
### Abstract

We prove that the energy dissipation property of gradient flows extends to semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the p -parabolic extension does not increase the p -norm of the gradient when p > 2 {p>2} . We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients. These are the first regularity results for vertical maximal functions without convolution structure.

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October 4, 2023
### Abstract

In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus–Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments.

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October 4, 2023
### Abstract

We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 𝒜 {{\mathcal{A}}} -superharmonic functions with nonlinear operator 𝒜 : Ω × ℝ n → ℝ n {{\mathcal{A}}:\Omega\times{\mathbb{R}^{n}}\to{\mathbb{R}^{n}}} having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space.

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August 31, 2023
### Abstract

In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies E ε {E_{\varepsilon}} defined in a randomly perforated domain, as ε goes to zero. The functionals E ε {E_{\varepsilon}} model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step, we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of E ε {E_{\varepsilon}} . Then the effective volume and surface energy densities are identified by means of a careful limit procedure.

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August 25, 2023
### Abstract

The model introduced in [45] in the framework of the theory on stress-driven rearrangement instabilities (SDRI) [3, 43] for the morphology of crystalline materials under stress is considered. As in [45] and in agreement with the models in [50, 55], a mismatch strain, rather than a Dirichlet condition as in [19], is included into the analysis to represent the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number m of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities [6, 35, 34, 45]. Due to the lack of compactness and lower semicontinuity for the sequences of m -minimizers, i.e., minimizers among configurations with at most m connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of m -minimizers’ energies to the energy infimum as m → ∞ {m\to\infty} . Finally, regularity properties for the morphology satisfied by every minimizer are established.

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August 25, 2023
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In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 {n-3} . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth.

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August 25, 2023
### Abstract

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ( ( | D u | - ν ) + p - 1 D u | D u | ) = f in Ω T = Ω × ( 0 , T ) , u_{t}-\operatorname{div}\Bigl{(}(\lvert Du\rvert-\nu)_{+}^{p-1}\frac{Du}{% \lvert Du\rvert}\Bigr{)}=f\quad\text{in }\Omega_{T}=\Omega\times(0,T), where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} for n ≥ 2 {n\geq 2} , p ≥ 2 {p\geq 2} , ν is a positive constant and ( ⋅ ) + {(\,\cdot\,)_{+}} stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t {u_{t}} . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

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Open Access
July 26, 2023
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Given ε 0 > 0 {{\varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {I\in\mathbb{N}\cup\{0\}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ ε 0 {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {M\looparrowright X} be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] {H\in[0,H_{0}]} with index at most I . For such M ↬ X {M\looparrowright X} , we prove a structure theorem which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M , where the norm of the second fundamental form takes on large local maximum values.

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July 25, 2023
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Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO ( M ) → M {\mathrm{SO}(M)\rightarrow M} of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o ( M ) ≅ SO ( M ) {\mathrm{Iso}_{o}(M)\cong\mathrm{SO}(M)} : A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

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July 25, 2023
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By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature.

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June 27, 2023
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We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.

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### Abstract

In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ( Q 0 ) := sup Q ⊂ Q 0 inf c ∈ ℝ 1 l ( Q ) β ∫ Q | u - c | 𝑑 ℋ ∞ β , \displaystyle\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}\vcentcolon=\sup_{Q\subset Q_{% 0}}\inf_{c\in\mathbb{R}}\frac{1}{l(Q)^{\beta}}\int_{Q}|u-c|\,d\mathcal{H}^{% \beta}_{\infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ( Q ) {l(Q)} is the length of the side of the cube Q , and ℋ ∞ β {\mathcal{H}^{\beta}_{\infty}} is the Hausdorff content. In the case β = d {\beta=d} we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ ( 0 , d ] {\beta\in(0,d]} one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C > 0 {c,C>0} such that ℋ ∞ β ( { x ∈ Q : | u ( x ) - c Q | > t } ) ≤ C l ( Q ) β exp ( - c t ∥ u ∥ BMO β ( Q 0 ) ) \displaystyle\mathcal{H}^{\beta}_{\infty}(\{x\in Q:|u(x)-c_{Q}|>t\})\leq Cl(Q)% ^{\beta}\exp\biggl{(}-\frac{ct}{\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}}\biggr{)} for every t > 0 {t>0} , u ∈ BMO β ( Q 0 ) {u\in\mathrm{BMO}^{\beta}(Q_{0})} , Q ⊂ Q 0 {Q\subset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}\in\mathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.

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June 1, 2023
### Abstract

We study minimisation problems in L ∞ {L^{\infty}} for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p {L^{p}} approximations as p → ∞ {p\to\infty} , we illustrate the existence of a special L ∞ {L^{\infty}} minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ {L^{\infty}} variational problem.

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May 6, 2023
### Abstract

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ( θ ) ϕ ( ρ ) α − 2 ϕ ( ρ ) 2 + | ∇ ¯ ρ | 2 , K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+\lvert\overline{\nabla}\rho\rvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 \alpha=n+1 ) in hyperbolic space, where f ~ \tilde{f} is a smooth positive function on S n \mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ( ρ ) = sinh ρ \phi(\rho)=\sinh\rho and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 \alpha\geq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 \mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2<\alpha\leq n+1 under the evenness assumption of f ~ \tilde{f} and prove the existence of solutions to the above equations.

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May 3, 2023
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We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.

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April 27, 2023
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In the paper we prove the convergence of viscosity solutions u λ {u_{\lambda}} as λ → 0 + {\lambda\rightarrow 0_{+}} for the parametrized degenerate viscous Hamilton–Jacobi equation H ( x , d x u , λ u ) = α ( x ) Δ u , α ( x ) ≥ 0 , x ∈ 𝕋 n H(x,d_{x}u,\lambda u)=\alpha(x)\Delta u,\quad\alpha(x)\geq 0,\quad x\in\mathbb% {T}^{n} under suitable convex and monotonic conditions on H : T * M × ℝ → ℝ {H:T^{*}M\times\mathbb{R}\rightarrow\mathbb{R}} . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation H ( x , d x u , 0 ) = α ( x ) Δ u . H(x,d_{x}u,0)=\alpha(x)\Delta u.

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March 31, 2023
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In this paper, we consider the first eigenvalue λ 1 ( Ω ) {\lambda_{1}(\Omega)} of the Grushin operator Δ G := Δ x 1 + | x 1 | 2 s Δ x 2 {\Delta_{G}:=\Delta_{x_{1}}+\lvert x_{1}\rvert^{2s}\Delta_{x_{2}}} with Dirichlet boundary conditions on a bounded domain Ω of ℝ d = ℝ d 1 + d 2 {\mathbb{R}^{d}=\mathbb{R}^{d_{1}+d_{2}}} . We prove that λ 1 ( Ω ) {\lambda_{1}(\Omega)} admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in ℝ d 1 {\mathbb{R}^{d_{1}}} and a set in ℝ d 2 {\mathbb{R}^{d_{2}}} , and that the minimizer is the product of two balls Ω 1 * ⊆ ℝ d 1 {\Omega^{*}_{1}\subseteq\mathbb{R}^{d_{1}}} and Ω 2 * ⊆ ℝ d 2 {\Omega_{2}^{*}\subseteq\mathbb{R}^{d_{2}}} . Moreover, we provide a lower bound for | Ω 1 * | {\lvert\Omega^{*}_{1}\rvert} and for λ 1 ( Ω 1 * × Ω 2 * ) {\lambda_{1}(\Omega_{1}^{*}\times\Omega_{2}^{*})} . Finally, we consider the limiting problem as s tends to 0 and to + ∞ {+\infty} .

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March 31, 2023
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In this work, we analyze Merriman, Bence and Osher’s thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoğlu and Otto, we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.

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Open Access
March 31, 2023
### Abstract

In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n {{\mathbb{H}^{n}}} via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 {\mathbb{H}^{1}} under an assumption on the location of the singular set, and in ℍ n {\mathbb{H}^{n}} for n ≥ 2 {n\geq 2} in the proper class of horizontally umbilical hypersurfaces.

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February 28, 2023
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In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau’s problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau’s problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows.

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January 27, 2023
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The homogeneous causal action principle on a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler–Lagrange equations are derived and analyzed under suitable regularity assumptions.

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January 27, 2023
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In this paper, when studying the connection between the fractional convexity and the fractional p -Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ( x ) {u(x)} is so-called ( α , p ) {(\alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(\alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.

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January 27, 2023
### Abstract

Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M . By the Sobolev embedding theorem, we can write that there exist K , B > 0 {K,B>0} such that for any u ∈ H 1 ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ∥ ∇ A u ∥ L 2 2 + B ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq K\|\nabla_{A}u\|_{L^{2}}^{2}+B\|u\|_{L^{1}}^{2} where H 1 ( M ) {H^{1}(M)} is the standard Sobolev space of functions in L 2 {L^{2}} with one derivative in L 2 {L^{2}} , | ∇ A u | 2 = A ( ∇ u , ∇ u ) {|\nabla_{A}u|^{2}=A(\nabla u,\nabla u)} and 2 ⋆ {2^{\star}} is the critical Sobolev exponent for H 1 {H^{1}} . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality.

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December 6, 2022
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We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.

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Open Access
November 24, 2022
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In this paper, we introduce the L p {L_{p}} q -torsional measure for p ∈ ℝ {p\in\mathbb{R}} and q > 1 {q>1} by the L p {L_{p}} variational formula for the q -torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p {L_{p}} Minkowski problem with respect to the q -torsional rigidity for discrete measures and general measures when 0 < p < 1 {0<p<1} and q > 1 {q>1} .

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November 11, 2022
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We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ( Ω ) {E_{f}(\Omega)} of the Laplacian in the domain Ω or the first eigenvalue λ 1 ( Ω ) {\lambda_{1}(\Omega)} of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ( Ω ) {E_{f}(\Omega)} or λ 1 ( Ω ) {\lambda_{1}(\Omega)} when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ( Ω ) {E_{f}(\Omega)} and λ 1 ( Ω ) {\lambda_{1}(\Omega)} among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ( Ω ) {E_{f}(\Omega)} and λ 1 ( Ω ) {\lambda_{1}(\Omega)} interesting for themselves.

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November 11, 2022
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We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.

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Open Access
October 25, 2022
### Abstract

We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a Γ-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in L p {L^{p}} , which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, k -th order, homogeneous differential operators with constant coefficients and constant rank.

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Open Access
October 11, 2022
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We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions. Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.

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October 8, 2022
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We consider the volume constrained fractional mean curvature flow of a nearly spherical set and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.

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Open Access
September 30, 2022
### Abstract

Eigenvalue problems for the p -Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q {L^{q}} - or L ∞ {L^{\infty}} -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.

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August 30, 2022
### Abstract

In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ { u > ϕ } {\partial\{u>\phi\}} to the obstacle problem of the porous medium equation, u t = Δ u m {u_{t}=\Delta u^{m}} ( m > 1 {m>1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ { u > ϕ } {\partial\{u>\phi\}} and ∂ { u > 0 } {\partial\{u>0\}} , we consider two cases on the initial data which make the free boundary ∂ { u > ϕ } {\partial\{u>\phi\}} separate from the free boundary ∂ { u > 0 } {\partial\{u>0\}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ { u > ϕ } {\partial\{u>\phi\}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.

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August 30, 2022
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We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ( u ) ≔ ∫ Ω f ( x , D u ( x ) ) 𝑑 x , \mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx, where the density f ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {x\in\Omega\subset\mathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ( x , z ) ⩽ L ( 1 + | z | q ) , \lvert z\rvert^{p}\leqslant f(x,z)\leqslant L(1+\lvert z\rvert^{q}), where 1 < p < q < p ( n + α n ) {1<p<q<p(\frac{n+\alpha}{n})} , and it can be approximated from below by suitable densities f k {f_{k}} .

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Open Access
July 23, 2022
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This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established in [J. Fischer, S. Hensel, T. Laux and T. M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, preprint 2020, https://arxiv.org/abs/2003.05478], under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

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Open Access
July 22, 2022
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We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p -Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equations. In the semilinear case p = 2 {p=2} the results are already new since they include new admissible nonlinearities.

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June 30, 2022
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We study local regularity properties of local minimizers of scalar integral functionals of the form ℱ [ u ] := ∫ Ω F ( ∇ u ) - f u d x \mathcal{F}[u]:=\int_{\Omega}F(\nabla u)-fu\,dx where the convex integrand F satisfies controlled ( p , q ) {(p,q)} -growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term f and improved assumptions on the growth conditions on F with respect to the existing literature. Along the way, we establish an L ∞ {L^{\infty}} - L 2 {L^{2}} -estimate for solutions of linear uniformly elliptic equations in divergence form, which is optimal with respect to the ellipticity ratio of the coefficients.

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June 29, 2022
### Abstract

Let Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n N → ℝ {F:\Omega\times\mathbb{R}^{nN}\to\mathbb{R}} , F ( x , p ) {F(x,p)} , be C 0 {C^{0}} , differentiable with respect to p , and with F p := D p F {F_{p}:=D_{p}F} continuous on Ω × ℝ n N {\Omega\times\mathbb{R}^{nN}} and F strictly convex in p . Consider an n N × n N {nN\times nN} matrix A = ( A α β i j ) ∈ C 0 ( Ω ) {A=(A^{{ij}}_{\alpha\beta})\in C^{0}(\Omega)} satisfying (0.1) A α β i j ( x ) ξ α i ξ β j = A β α j i ( x ) ξ α i ξ β j ≥ λ | ξ | 2 , λ > 0 . A^{ij}_{\alpha\beta}(x)\xi^{i}_{\alpha}\xi^{j}_{\beta}=A^{ji}_{\beta\alpha}(x)% \xi^{i}_{\alpha}\xi^{j}_{\beta}\geq\lambda\lvert\xi\rvert^{2},\quad\lambda>0. Suppose that (0.2) lim | p | → ∞ 1 | p | ( D p F ( x , p ) - A ( x ) p ) = 0 , \displaystyle\lim_{\lvert p\rvert\to\infty}\frac{1}{\lvert p\rvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 | p | 2 ≤ F ( x , p ) ≤ C 0 ( 1 + | p | 2 ) , \displaystyle{-}C_{0}+c_{0}\lvert p\rvert^{2}\leq F(x,p)\leq C_{0}(1+\lvert p% \rvert^{2}), (0.4) | F p ( x , p ) - F p ( x , q ) | ≤ C 0 | p - q | , \displaystyle\lvert F_{p}(x,p)-F_{p}(x,q)\rvert\leq C_{0}\lvert p-q\rvert, (0.5) 〈 F p ( x , p ) - F p ( x , q ) , p - q 〉 ≥ c 0 | p - q | 2 \displaystyle\langle F_{p}(x,p)-F_{p}(x,q),p-q\rangle\geq c_{0}\lvert p-q% \rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ( u ) := ∫ Ω F ( x , D u ( x ) ) 𝑑 x + ∫ Ω G ( x , u ) 𝑑 x , J(u):=\int_{\Omega}F(x,Du(x))\,dx+\int_{\Omega}G(x,u)\,dx, where G ( x , ⋅ ) ∈ C 1 ( ℝ N ) {G(x,\cdot\,)\in C^{1}(\mathbb{R}^{N})} for each x ∈ Ω {x\in\Omega} , G ( ⋅ , u ) {G(\,\cdot\,,u)} is measurable for each u ∈ ℝ N {u\in\mathbb{R}^{N}} , and (0.7) | G u ( x , u ) | ≤ C 0 ( 1 + | u | s ) \lvert G_{u}(x,u)\rvert\leq C_{0}(1+\lvert u\rvert^{s}) with s < n + 2 n - 2 {s<\frac{n+2}{n-2}} . Under these conditions, we shall show that if n > 2 {n>2} , then any weak solution u ∈ W 1 , 2 ( Ω , ℝ N ) {u\in W^{1,2}(\Omega,\mathbb{R}^{N})} of the Euler equations of J , i.e. ∑ α ∂ ∂ x α F p α i ( x , D u ) = G u i ( x , u ) , i = 1 , … , N , \sum_{\alpha}\frac{\partial}{\partial x^{\alpha}}F_{p^{i}_{\alpha}}(x,Du)=G_{u% ^{i}}(x,u),\quad i=1,\ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.

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June 29, 2022
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We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L 1 {L^{1}} initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss–Green formula.

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May 31, 2022
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In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings.