1 Introduction and main results
In this work, we consider the following degenerate (or singular) elliptic equations of p-Laplacian type:
defined in an open and bounded set and for . The modulus of ellipticity of the p-Laplace operator is . When , the modulus vanishes whenever , and the equation is called degenerate at those points where that occurs. On the other hand, for , the modulus becomes infinite when , and the equation is called singular at those points. Observe that the case is just the linear case and corresponds to the Laplace operator.
Different notions of solutions have been formulated for equation (1.1). We are interested in the relation between Sobolev weak solutions and viscosity solutions. For the homogeneous p-Laplace equation, this relation has already been studied by Juutinen, Lindqvist and Manfredi in , via the notion of p-harmonic, p-subharmonic and p-superharmonic functions. Roughly speaking, a p-harmonic function is a continuous function which solves, weakly, the homogeneous p-Laplace equation, and a p-superharmonic (p-subharmonic) function is a lower (upper) semicontinuous function that admits comparison with p-harmonic functions from below (above).
In , Juutinen, Lindqvist and Manfredi showed that the notion of p-harmonic solution is equivalent to the notion of viscosity solution. Moreover, it was shown in  that locally bounded p-harmonic functions are weak solutions. Conversely, every weak solution to the homogeneous p-Laplace equation has a representative which is lower semicontinuous and it is p-harmonic. We refer the interested reader to  for further details. In this way, there is an equivalence between the notion of weak and viscosity solutions for the homogeneous framework. It is worth to mention that a different and simpler proof of this equivalence was stated by Julin and Juutinen in  by using inf and sup convolutions. In turn, this reasoning was extended in  to more general second-order differential equations.
For the non-homogeneous case, the notion of p-harmonic functions is lost and we need to study directly the link between viscosity and Sobolev weak solutions. In , the authors showed that viscosity solutions of (1.1) are weak solutions in the case where f is continuous and depends only on x.
Our main goal in the present manuscript is to prove the equivalence of these two notions of solutions for the general structure (1.1). The implication that viscosity solutions are weak solutions is partially based on the work , but the non-homogeneous nature of the equation under consideration requires some extra effort to deal with the lower-order term.
On the other hand, the converse statement relies on comparison principles for weak solutions. To the best of our knowledge, the available comparison results for the full case require additional limitations in the degenerate case which do not appear in the singular context (compare Theorem A.1 and Theorem A.2). Moreover, we believe that the assumption that weak subsolutions and weak supersolutions belong to or to the Sobolev space in order to have comparison is not a strong limitation since we are interested in the equivalence of weak and viscosity solutions, and for weak solutions the -regularity holds (see [4, 16]). Finally, in the quasi-linear case there is no need to impose higher regularity than on the solutions. We refer the reader to  for a survey of maximum principles and comparison results for general structures in divergence form.
Finally, we stress that the equivalence between weak and viscosity solutions may be used to prove relevant properties on the solutions. As an example, in , Juutinen and Lindqvist prove a Radó’s-type theorem for p-harmonic functions. Roughly speaking, they state that if a function u solves, weakly, the homogeneous p-Laplace equation in the complement of the set where u vanishes, then it is a solution in the whole set. It is an open problem to obtain a similar result for equations like (1.1). We shall return to this issue in a subsequent paper.
We recall that the p-Laplace operator is defined as
Let us state the different type of solutions to (1.1) we will manage.
Definition 1.1 (Sobolev weak solution).
A function is a weak supersolution to (1.1) if
for all non-negative . On the other hand, u is a weak subsolution if is a weak supersolution of the equation . We call u a weak solution if it is both a weak subsolution and a weak supersolution to (1.1).
A lower semicontinuous function is a viscosity supersolution to (1.1) if and for every such that , and for all , there holds
A function u is a viscosity subsolution if is a viscosity supersolution to the equation , and it is a viscosity solution if it is both a viscosity sub- and supersolution.
Notice that condition (1.2) is established this way to avoid the problems derived from having in the case . If , this condition can be simply replaced by
We now list the main contributions of our work. The results are stated for supersolutions, but they hold for subsolutions as well.
Let . Assume that is uniformly continuous in , non-increasing in s, and satisfies the growth condition
A converse of Theorem 1.4 is given below.
Assume that is continuous in , non-increasing in s, and locally Lipschitz continuous with respect to η. Hence we have the following:
According to recent results (see ), it is possible to weak the locally Lipschitz assumption in Theorem 1.5 when f takes some particular forms or it satisfies extra convexity and coercivity assumptions. For instance, as a consequence of the results in , if
where is decreasing and .
In view of the available regularity theory for weak solutions of (1.1), we have the following equivalence.
Let . Assume that is uniformly continuous, locally Lipschitz in η, non-increasing in s, and satisfies the growth condition (1.3). Additionally, assume that for and when . Then u is a weak solution to (1.1) if and only if it is a viscosity solution to (1.1).
We point out that, in the degenerate case, it is possible to remove the assumption by imposing the non-vanishing of the gradient of the weak solution in the whole Ω. This is a straightforward consequence of Theorem 1.5 (iii).
In the particular case where f does not depend on η, we have the following converse to Theorem 1.4 which does not require the locally Lipschitz regularity of the solutions.
Let us briefly discuss the above hypotheses on f. Firstly, assuming that f is non-increasing and introducing the operator
we derive that F is proper, that is, F is non-increasing in and non-decreasing in s, which is a standard and useful assumption in the theory of viscosity solutions . For instance, it allows to get the equivalence between classical solutions ( functions which satisfy the equations pointwise) and viscosity solutions. On the other hand, the growth property (1.3) implies the -regularity of weak solutions to (1.1) (see [4, 17, 16]). Moreover, under a regular Dirichlet boundary condition , the -regularity up to the boundary of weak solutions follows. For further details, see the reference . Finally, the extra assumption appearing in Theorem 1.5 in the degenerate case is used to remove critical sets of points of the weak solution (see reference ). Hence, it allows the application of comparison results without assuming the non-vanishing of the gradients. We point out that other properties of , as more regularity on s and η and convexity-like conditions, may be employed to ensure comparison for weak solutions. We refer the reader to  and the references therein for more details.
It is worth mentioning that many equations appearing in the literature have the structure of (1.1) with the lower-order term satisfying the above assumptions on f. We refer the reader to [14, 15, 3, 2] and the references therein for examples of such f.
The paper is organized as follows: In Section 2, we provide some preliminary results concerning properties of infimal convolutions (which will be the main tool in the proof of Theorem 1.4) and a convergence result. In addition, we prove a Caccioppoli-type estimate that will provide important uniform bounds, fundamental when using approximation arguments. This result is interesting in itself.
Section 3 contains the proof of the main result of the paper, Theorem 1.4, that states under which conditions on the non-homogeneous function f in (1.1) viscosity solutions are actually weak solutions. This proof is divided into two major cases: the singular and the degenerate scenario, thus, although both cases rely on the same idea, different approximations and estimates are needed depending on the range of p.
In Section 4, we prove the reverse statement, that is, weak solutions of (1.1) are viscosity solutions. This result is based on comparison arguments, and this will determine the conditions we will need to impose on f. Finally, in Appendix A we give, for the sake of completeness, precise references and state the comparison results that we use in Section 4.
2 Preliminary results
2.1 Infimal convolution
Let us define the infimal convolution of a function u as
where and .
We recall some useful properties of . Let be bounded and lower semicontinuous in Ω. It is well known that is an increasing sequence of semiconcave functions in Ω, which converges pointwise to u. Hence, is locally Lipschitz and twice differentiable a.e. in Ω. Moreover, it is possible to write
The next lemma is the counterpart of [6, Lemma A.1 (iii)] for our setting.
Suppose that is bounded and lower semicontinuous in Ω. Let be continuous in and non-increasing in s. If u is a viscosity supersolution to
in Ω for , then is a viscosity supersolution to
in , where
We start by noticing that
Let us see first that for every , the function
is a viscosity supersolution to in . Indeed, let and so that
We assume that for all if . Making , and
we derive that has a local minimum at , and indeed . Since u is a viscosity supersolution to (2.2), there follows
where we have used that f is non-increasing in the second variable. Let us see now that, since is an infimum of supersolutions, it is itself a supersolution (observe that is continuous, since it is locally Lipschitz). Let and so that
Again, for all in the singular scenario. Moreover, we may assume that the minimum is strict. For each n, there exists such that
Let be a sequence of points in so that
for all , i.e., has a minimum in at . Up to a subsequence, as . Furthermore, by (2.5),
Taking liminf and using the lower semicontinuity of , we derive
Since the minimum in (2.4) is strict, we must have . Moreover, taking
in (2.3), we have
Since f is non-increasing with respect to the second variable, by (2.6) we obtain
As , there holds
for some . Therefore,
and we conclude that is a viscosity supersolution of
as desired. ∎
The next lemma states the weak convergence of the lower-order terms in the particular situation of infimal convolutions.
Let be a uniformly continuous function, which satisfies the growth condition (1.3). Assume that is locally bounded and lower semicontinuous in Ω. For each define as in (2.1) and as in Lemma 2.1. Then, if converges to in , the following holds:
for every non-negative .
Let and denote spt. Consider small enough so that
where . Since f is uniformly continuous in , for every there exists such that
Choose so that for every . Thus, from the previous inequality we get
for every and . In particular,
Hence we arrive at the estimate
On the other hand, due to the continuity of f and the convergences of and ,
Since , u belong to , there exists a uniform constant so that
Thus, in view of the growth estimate on f and the continuity of γ, we have, for an appropriate positive constant C,
Since , Hölder’s inequality and the strong convergence of imply
2.2 A Caccioppoli’s estimate
In the next lemma we provide a Caccioppoli’s estimate for the -norm of the gradients of weak solutions.
where , and .
Let and be as in the lemma. Consider the test function
which shows that . Hence, Young’s inequality
where q and are conjugate exponents, implies that the first integral on the right-hand side of (2.9) may be bounded by
Moreover, by (1.3) we have
for all x in the support of ξ, where . Therefore, the second integral in (2.9) is estimated from above by
where is a positive constant. The assumption and Young’s inequality yield
Taking , we derive Caccioppoli’s estimate. ∎
3 Proof of Theorem 1.4
3.1 Degenerate case
We begin with the range .
Proof of Theorem 1.4.
Let be the infimal convolution defined in (2.1) with . Then
a.e. in . Furthermore,
for all non-negative test functions ψ (see the proof of [6, Theorem 3.1]). Hence, we derive
for all non-negative test functions ψ and all . We claim that, as , there holds
To prove the claim, observe first that Caccioppoli’s estimate allows us to conclude that
converges weakly in . Indeed, for any compact set , choose an open set containing K and a non-negative test function so that
and in K. Then
Moreover, since is an increasing sequence and converges pointwise to u in Ω, we have
Hence converges weakly in , and converges weakly in . Since converges pointwise to u, we derive that and converges weakly in to u.
More can be said: converges strongly in to . Indeed, take
where θ is a non-negative smooth test function compactly supported in Ω. From
By the weak convergence of to u in , the last integral in (3.3) tends to 0 as . The left-hand side is given by
The second integral in (3.4) is estimated in absolute value by
which tends to 0 as . Moreover, since
with , does not depend on ε, it also holds that
where we have used the fact that the integrand is always non-negative. Therefore, using the inequality
and, in turn, the strong convergence of the gradients and Lemma 2.2 gives
This ends the proof of the claim and we deduce that u is a weak supersolution. ∎
3.2 The singular case:
Consider now the infimal convolution given in (2.1) choosing , i.e.,
Notice that for .
We need the following auxiliary result, which is an adaptation of [6, Lemma 4.3].
Suppose that u is a bounded viscosity supersolution to (1.1). If there is such that is differentiable at and , then .
From [6, Lemma 4.3] we know that , and hence
which satisfies , and
in view of . Since , for , for all , and u is a viscosity supersolution to (1.1),
Noticing that and , by (3.7) we conclude
as desired. ∎
We can prove now Theorem 1.4 in the case .
Proof of Theorem 1.4.
a.e. in . Performing the same approximation argument as in the proof of [6, Theorem 4.1], we reach that
for every , . Therefore and since by Lemma 3.1 we know in the set , we get
Repeating the proof for the case (and noticing that Lemma 2.3 works for every ), we obtain the uniform boundedness of in and
for any compact set , and from here the convergence
Using Hölder’s inequality and the vector inequality (see [4, Chapter I])
with and , we obtain
Proof of Theorem 1.5 (i).
Let be a weak supersolution to (1.1). To reach a contradiction, assume that u is not a viscosity supersolution. By assumption, there exist and so that for all ,
Moreover, by the -regularity of u, we may assume that u is continuous in Ω. Thus, the map
is continuous in Ω, and (4.1) yields
Hence, there exists some so that
Then by (4.1) we have . Consider
By (4.2), is a weak subsolution to
in , where . Observe that is continuous in and locally Lipschitz in η. Moreover, in the weak sense, we have
Proof of Theorem 1.5 (ii).
For a given weak supersolution , by following the lines above and appealing to the comparison Theorem A.1, we can show that u is a viscosity supersolution in the non-critical set
By [8, Corollary 4.4], which holds true for sub- and supersolutions, u is a viscosity supersolution in the whole set Ω. ∎
Proof of Theorem 1.5 (iii).
The proof follows similarly just by using the assumption in Ω together with the comparison Theorem A.1. ∎
Observe that the -regularity of u is only needed to apply the comparison principles. In the rest of the proof, the continuity of u would be enough.
A.1 Comparison principles for weak solutions
In this section, we provide the comparison principles for weak solutions of (1.1) that we use in the proof of Theorem 1.5. As we pointed out in Section 1, other comparison results may be employed (see ).
The first one is contained in [15, Corollary 3.6.3].
Assume that is continuous in , non-increasing in s, and locally Lipschitz continuous with respect to η in . Let be a weak supersolution and let be a weak subsolution to (1.1) in Ω for . Assume that in Ω. If on , then in Ω.
In the singular case, the assumptions on the gradients may be removed. See [15, Corollary 3.5.2].
Let . Assume that is continuous in , non-increasing in s, and that it is locally Lipschitz continuous in η on compact subsets of its variables. Then if is a weak supersolution and if is a weak subsolution to (1.1) in Ω so that on , then in Ω.
Finally, in the case where f does not depend on η, we have the following result (see [15, Corollary 3.4.2]).
Suppose that is continuous in and non-increasing in s. Let be a supersolution and a subsolution so that on . Then in Ω.
where the operator is assumed to be continuous in , continuously differentiable with respect to s and η for all s and all , and elliptic in the sense that is positive definite in . In the particular case of the p-Laplace operators
all of the assumptions above are satisfied. The positive definiteness of is a consequence of
for a positive constant c. Finally, observe that A is uniformly elliptic for if . This allows the improved comparison result in Theorem A.2.
The authors would like to thank the anonymous referee for her/his comments.
M. Cuesta and P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (2000), no. 4–6, 721–746. Google Scholar
E. DiBenedetto, -local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850. Google Scholar
J. Heinonen, T. Kilpelainen and O. Martio, Non-Linear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Mineola, 2006. Google Scholar
V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Comm. Partial Differential Equations 37 (2012), no. 5, 934–946. CrossrefGoogle Scholar
P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal. 33 (2001), no. 3, 699–717. CrossrefGoogle Scholar
N. Katzourakis, Nonsmooth convex functionals and feeble viscosity solutions of singular Euler–Lagrange equations, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 275–298. CrossrefWeb of ScienceGoogle Scholar
P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67–70. Google Scholar
S. Merchán, L. Montoro and B. Sciunzi, On the Harnack inequality for quasilinear elliptic equations with a first order term, preprint (2016), https://arxiv.org/abs/1601.03863.
P. Pucci and J. Serrin, The Maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Boston, 2007. Google Scholar
About the article
Published Online: 2017-06-04
The first author was supported by the grant FONDECYT Postdoctorado 2016, No. 3160077. The second author was partially supported by CONICET and grant PICT 2015-1701 AGENCIA.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 468–481, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0005.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0