Half-space Gaussian symmetrization: applications to semilinear elliptic problems

We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definitions of solutions in order to prove the existence and uniqueness of a solution in R without growth restrictions at infinity. A comparison result in terms of the halfspace Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.


Introduction
In this paper we focus our attention on a class of semilinear elliptic Dirichlet problems, whose prototype is where c > , p > , Ω is an open subset of R N not necessary bounded, φ(x) = φ N (x) := ( π) − N exp − |x| is the density of standard N−dimensional Gauss measure γ and the datum f belongs to a suitable Zygmund space. A more general di usion operator is in fact considered in all this paper.
Problem (1.1) is related to Ornstein-Uhlenbeck operator Lu := u − x · ∇u and our approach allows us to consider an extra semilinear zero order term c |u| p− u, Ω = R N and a weak assumption on the summability of datum. Notice that we can formally write div(∇u φ(x)) = uφ(x)+∇φ(x)·∇u which justi es the multiplicative role of φ(x) in the equation of (1.1). The idea of "symmetrizing the operator" −∆u(x) + x · ∇u in order to solve the drift equation comes back from a pioneering paper [27] by Kolmogorov in 1937 for c = . For some recent survey in this direction see [32]. It is well-known the above di usion operator with a drift is related to the stochastic Ornstein-Uhlenbeck process with applications in nancial mathematics and in physical sciences (a model for the velocity of a massive Brownian particle under the in uence of friction ). This is sometimes also written in terms of a Langevin ordinary di erential equation with noise (see, e.g. [31]). We remark that if φ ≡ problem (1.1) was largely considered in the literature (see, e.g. [9] when Ω is bounded and [37] and [19] when Ω is unbounded and p > ).
In the weighted case, when p ≤ and γ(Ω) < , Lax-Milgram Theorem guarantees the existence of a solution for problem (1.1) u ∈ H (Ω, γ) (the weighted Sobolev space) once we assume f belongs to its dual. For example we can require f ∈ L log L − (Ω, γ), a functional space which we recalled in Section 2, where other preliminary notions will also be collected.
If we consider Ω = R N one of the di culties that arises when solving (1.1) is due to lack of a Poincaré inequality. As a consequence we have to consider the Banach space H (R N , γ) equipped with the norm u L (R N ,γ) + ∇u L (R N ,γ) . In the linear case, p = , the existence and uniqueness of a weak solution u ∈ H (R N , γ) to (1.1) follows again from Lax-Milgram Theorem and we also get the correct growth condition on f (x) as |x| → +∞ (see, e.g. the exposition made in [30]).
The superlinear case p > is di erent. In Section 3 we shall prove the existence and uniqueness giving a suitable notion of weak solution for the case of Ω = R N and p > . We point out that in the superlinear case when γ(Ω) < the existence and uniqueness of a weak solution can be obtained through an easy adaptation of the results of Brezis and Browder [9]. In order to consider Ω = R N we follow an idea of [19], giving an alternative proof and enlarging the applications of the pioneering result on superlinear problems by Brezis [8]. Thanks to the assumption p > we get some a priori estimates on any given half-space allowing us to obtain a general existence and uniqueness result without any growth condition at in nity on the datum f (and so with less summability than f in L log L − (R N , γ)).
In a second part of the paper (Sections 4 and 5) we deal with comparison results in terms of the half-space Gaussian symmetrization of solutions of (1.1) and its generalizations. We point out that when Ω is bounded the usual radially symmetrization method, when applied to an elliptic operator with a drift term as (1.2) modi es drastically the drift term in the symmetrized equation (see, e.g., [34]). In contrast to that, the half-space Gaussian symmetrization method allows to preserve the more important facts of the drift term (see Remark 11) as well as to deal with an unbounded domain. We recall that in [3] the authors compare the solution u to problem ( . ) , when γ(Ω) < , with the solution v to a simpler problem, called the half-space Gaussian symmetrized problem, without zero order term de ned in the half-space Ω = {(x , ..., x N ) ∈ R N : x > ω} with ω such that γ(Ω ) = γ(Ω) and with a datum depending only on the rst variable. Here we are interested to prove some comparisons in term of the solution to symmetrized problem keeping also a nonlinear zero order term. As in the unweighted case (see e.g. [16], [18]) we are able to obtain some integral estimates, that imply a comparison between Lebesgue norms. Other comparison results related to Gauss measure are contained in [20,21], [12] for the parabolic case and [25] in the non-local case. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations p ∈ ( , ) when the datum f possibly vanishes on a positively measured subset of Ω.
Finally in Section 5 we generalize our results to problems with more general zero order terms b(u), not necessary of power type as in previous Sections.

Preliminaries
Let γ be the N-dimensional Gauss measure on R N de ned by In what follows we will set φ N (x) = φ(x) for simplicity.
It is well-known that an isoperimetric inequality for Gauss measure holds (see e.g. [11]): among all measurable sets of R N with prescribed Gauss measure, the half-spaces take the smallest perimeter (in the sense of this measure). In particular, the perimeter of a (N − )−recti able set E of R N with respect to the Gauss measure is de ned as where H N− denotes the (N − )−dimensional Hausdor measure. The following isoperimetric estimate (see e.g. [11]) the − dimensional Gauss measure of line (λ, ∞). Now we introduce the notion of rearrangement with respect to Gauss measure (see e.g. [22]). Here the balls of Schwartz symmetrization is replaced by half-spaces If u is a measurable function in Ω, we denote by u the decreasing rearrangement of u with respect to Gauss measure, i.e.
is the distribution function of u with respect to the Gauss measure. Moreover the rearrangement with respect to Gauss measure of u is de ned as where Ω * := Hω with ω such that γ Ω * = γ (Ω) (2.6) and Φ is de ned in (2.2). By de nition u * is a function which depend only on the rst variable and its level sets are half-spaces. Moreover u, u and u * have the same distribution function. If u (x) , v (x) are measurable functions an Hardy-Littlewood inequality hods: For general results about the properties of rearrangement with respect to a positive measure see, for example, [13]. We recall that for every open set Ω ⊆ R N the weighted Lebesgue space L p (Ω, γ) with p ≥ is the space of measurable functions u such that Moreover, as usual, H (Ω, γ) states for the weighted Sobolev space of functions u such that u, |∇u| ∈ L (Ω, γ) equipped with the norm u L (Ω,γ) + ∇u L (Ω,γ) . Finally we denote by H (Ω, γ) the closure of C ∞ (Ω) under the norm ∇u L (Ω,γ) . We remark that a Poincaré inequality holds only when γ(Ω) < : for every u ∈ H (Ω, γ), where C P is a positive constant depending on Ω.
The Sobolev space H (Ω, γ) is continuously embedded in the Zygmund space L (log L) (Ω, γ) (see [26], [24], [23] and references therein). We recall that given ≤ p < ∞ and −∞ < α < +∞, a measurable function u belongs to the Zygmund space L p (log L) α (Ω, γ) if Then, it is well-known that there exists a constant C S depending on Ω such that for all u ∈ H (Ω, γ). This explain why Zygmund spaces are the natural spaces for the data of problems as (1.1). We observe that these spaces give a re nement of the usual Lebesgue spaces. Indeed by de nition (2.9) the space L p (log L) (Ω, γ) = L p (Ω, γ). For the de nition and properties of the classical Zygmund space we refer to [5].
When Ω = R N we explicitly underline that inequality (2.10) holds by replacing the norm of the gradient by the norm of H (R N , γ).

Existence and uniqueness of solutions for the superlinear problem in R N without growing conditions
In the present section we focus our attention to existence and uniqueness of solutions to problem (1.1) when Ω = R N . Precisely we consider the more general second order elliptic problem As recalled in the introduction, to deal with Ω = R N we will need a suitable de nition of weak solution. We refer to [8,19] for unweighted case φ(x) ≡ .
We introduce the natural energy space for the linear problem (for the case p = ) where Hω is de ned in (2.3). We stress that for a given function f such that f ∈ L (log L) − (Hω , γ) for any Hω the integrals Hω fψ dγ are well-de ned for any ψ ∈ V(R N ) (even if f is not necessarily in the dual space of H (R N , γ)).
In this section we will assume the following structural conditions: De nition 1. A function u ∈ V(R N ) is a weak solution to problem ( . ) if c |u| p− u ∈ L (Hω , γ) for any ω ∈ R and for every ψ ∈ H (Hω , γ) ∩ L ∞ (Hω) with support contained in Hω for any ω ∈ R.
We stress that under our assumptions all terms in (3.2) are well-de ned. Obviously in De nition (1) we can consider any half-space not only the ones with boundary perpendicular to e . Let us x ω > and let us introduce for any ω ∈ R the auxiliary function Theorem 2. Let us suppose that (A1)-(A4) hold. Then, there exists a unique weak solution in the sense of Definition 1 to problem ( . ) such that c |u| p+ ∈ L (Hω , γ) for any ω ∈ R and (3.2) holds for ψ = uΘω for any ω ∈ R. Proof.
Step 1. Existence. For a given M ∈ N let us consider the following localized problem We will adapt Brezis-Browder's proof (see [9]) to prove the existence and uniqueness of a weak solution and Hω+ω |u| p+ dγ ≤K for some positive constants ε and δ that can be chosen later. Moreover we have for some positive constant ε ′ that can be chosen later. Sobolev inequality (2.10) and Young inequality allow us to obtain for some positive constants k , k , k , k , k . As before we get for some positive constant δ ′ that can be chosen later.
for some positive constant k and the last integral is nite if m > p+ p− . Moreover for some positive constant k , k . Using (3.9), (3.10) and (3.18) we obtain (3.7) and (3.8).
Using (3.7) and (3.8) we can conclude that u M is bounded in H (Hω+ω ,γ) and it follows that there exists u such that (up a subsequence) u M → u weakly in H (Hω , γ) for any ω ∈ R, weakly in L p+ (Hω , γ) for any ω ∈ R, strongly in L q (Hω , γ) for any ω ∈ R with q < p + and a.e. in Hω for any ω ∈ R. Using these convergences and the monotonicity of function G(s) = |s| p− s we can pass to the limit in (3.3) and we conclude.
Step 2. Uniqueness. Let u and u be two di erent weak solutions to Problem (3.1) such that c|u | p+ , c|u | p+ ∈ L (Hω , γ) for any ω ∈ R. We stress that they satisfy (3.2) with ψ = u Θω and ψ = u Θω For p > we have Γ(x) ≥ c a.e. in R N . Then Lemma 3 can be applied obtaining and for some positive constant K . Lebesgue's dominated convergence theorem allows Putting ω goes to −∞ we get ∇v = a.e. on R N by (3.19) and then v = a.e. on R N using (3.20). This is a contradiction and the uniqueness result follows.

Remark 4.
Arguing as in the proof of the uniqueness result it is possible to prove that if u is a weak supersolution and u is a weak solution to problem (3.1), then u ≤ u a.e. on R N .
and both topologies are equivalent.

Remark 7.
Since in Theorem 2 the existence (and uniqueness) of solutions is obtained without any decay condition on f it is natural to search about possible decay estimates of the solutions when |x| → +∞. The estimates obtained in [19] was extended to other di erent settings by several authors (see, e.g. [29] and its references).

Comparison results in terms of the half-space Gaussian symmetrization
In this section we will give results comparing a solution of problem of type (1.1) with the solution to a simpler problem de ned in an half-space having data depending only on one variable. We need starting with the case γ(Ω) < and we consider the following class of Dirichlet problems (4.1) The structural assumptions (instead of (A ) − (A )) are now the following: We recall that under assumption (A ′ ) a Poincaré inequality holds and that f ∈ L log L − / (Ω, γ) can be identi ed with an element in the dual space of H (Ω, γ) (see [4]). Then, under our assumptions, all terms in the corresponding notion of weak formulation are well-de ned using (2.10) and (2.8). Since γ(Ω) < , the existence of a weak solution to (4.1) comes easily by adapting the Brezis-Browder's proof ( [9]). Moreover, the uniqueness of solutions is standard.
The rst result of this section shows a suitable integral comparison between the solution u to problem ( . ) and the solution v to the following symmetrized problem where Ω is the half-space de ned in (2.6) and f is such that f = f , the rearrangement with respect to Gauss measure of f de ned in (2.5).
First of all, we prove that the solution to (4.2) coincides with its half-space Gaussian rearrangement.
Since f = f , then f ≥ . As a consequence the existence of a unique nonnegative weak solution is standard. We only detail the proof of where φ (x ) is the density of -dimensional Gauss measure and ω is such that γ(Ω * ) = γ(Ω). By uniqueness is the unique weak solution to Problem (4.2). Thus it remains to be proved the monotonicity. Since Suppose that Ψ(x ) < for some x ≥ ω and consider x ∈ [ω, +∞) such that Ψ(x ) = min [ω,+∞) Ψ(x ). It is obvious that Ψ(x ) < . We have that x > ω, otherwise it follows that In a similar way we show that there exists x ∈ (ω, x ) such that Ψ(x ) > . Indeed otherwise Proof. We argue as in [16], [18]. Let us de ne the functions u κ,t : Ω → R as for any xed t and κ > . Observing that u κ,t belongs to H (Ω, γ), and ∇u κ,t = χ {t<|u|≤t+κ} ∇u a.e. in Ω, function u κ,t can be chosen as test function in (18)  By Hardy-Littlewood inequality (2.7), we obtain Using (2.1) by standard arguments (see [33]) it follows that for t > .

Remark 10.
Under the same assumption of Theorem 9, it is well-known that we deduce also that u L rp (Ω,γ), ≤ v L rp (Ω ,γ) for any ≤ r ≤ ∞.

Remark 11. As mentioned in the Introduction, when Ω is bounded the usual radially symmetrization method, when applied to an elliptic operator with a drift term as (1.2), modi es drastically the drift term in the symmetrized equation. For instance, we can apply Theorem 1 of [34] to equation (1.2), which can be formulated in divergence form as
so that in the notation of [34] we must take b i (x) = x i and c(x) = −N. Then the corresponding symmetrized problem built in [34] is where Ω # is now a ball with the same volume than Ω, B = |x| L ∞ (Ω) , and f # is, for instance, the radially decreasing symmetric rearrangement of f . Notice that, in contrast with the "arti cial" rst order terms arising in problem (4.15), the half-space Gaussian symmetrization problem (4.2) preserves the same type of drift than the original problem (1.2) Using the De nition 1 the above comparison result can be extended to the case of Ω = R N and p > under the assumptions of the previous Section.

Theorem 12. Let Ω = R N , p > and the rest of conditions of Theorem 2. Let f be a nonnegative function and
admits a unique weak solution v. ii) let u and v be the nonnegative weak solution of (3.1) and of (4.16), respectively. Then, for any ε ∈ ( , ) where U, V, F and F are de ned as in Proof. Part i) is a consequence of Theorem 2 (see also [6]). To prove ii), as in the proof of Theorem 2, we rst consider u M , the solution of the corresponding localized problem (P −M ) on H −M , and v M the solution of the symmetrized problem  H (Hω+ω , γ) with ω ∈ R and ω > . It follows that there exists u and v such that (up a subsequence) u M → u and v M → v weakly in H (Hω) for any ω ∈ R, weakly in L p+ (Hω , γ) for any ω ∈ R, strongly in L q (Hω , γ) for any ω ∈ R with q < p + and a.e. in Hω for any ω ∈ R. Since the rearrangement application u → u is a contraction in L r (Hω , γ) for any r ≥ , we get that u M → u in L q (Hω , γ) for any ω ∈ R with q < p + and a.e. in Hω for any ω ∈ R (and weakly in L p+ (Hω , γ) for any ω ∈ R). On the other hand, by the Polya-Sezgo theorem which implies (thanks to the assumption p > : Lemma 3) that u M is bounded in H (Hω+ω , γ) and thus u M → u weakly in H (Hω , γ) for any ω ∈ R. In particular, if we de ne we get that for any M ∈ N. Moreover U M → U and V M → V strongly on L ∞ ( , − ε) for any ε ∈ ( , ) and thus we get the desired conclusion.

Remark 13.
Notice that we have proved the existence and, specially, the uniqueness of a solution of the problem when we assume only ≤ F(s) and F ∈ L loc ( , ), to be more precise F ∈ L ( , − ε) for any ε ∈ ( , ). For instance we could consider function of the type This type of questions is related with the study of removable singularities for quasilinear equations (see, e.g. Section 5.2 of Veron [36]). In this theory, usually it is assumed N ≥ . Remark 14. The above passing to the limit in M ∈ N also holds (for very di erent arguments, see Remark 6) when p ≤ once we assume that f ∈ L (R N , γ) is in the dual space of H (R N , γ) (which, essentially, corresponds to the case in which F ∈ L ( , )).
We end this Section with a qualitative property for the case p < which holds as an application of the above comparison Theorem. First of all we recall that this assumption allows the formation of a free boundary in the sense that if f = over some suitable large subset of Ω, with γ( x ∈ Ω : f (x) = ) = s f ∈ ( , γ(Ω)), then the solution u of (4.1) have compact support on Ω (i.e., Nu := x ∈ Ω : u(x) = , contained in x ∈ Ω : f (x) = , is not empty) (see, e.g. [15]). Thus γ( x ∈ Ω : u(x) = ) = τ, for some τ ∈ [ , s f ]. (4.17) Notice that in terms of the corresponding function U(s) it means that U attaints its maximum on a subinterval [γ(Ω) − τ, γ(Ω)]. Solutions V(s) of the symmetrized problem (4.9) may have also this property (once the data F(s) take its maximum on an interval [γ(Ω) − s F , γ(Ω)]). This is possible since the di usion operator of (4.9) becomes degenerate over the sets where V ′ (s) ≡ , because p < (see, Theorem 1.14 of [15]). The following result gives some estimates about the decaying (in measure) of u (t) near the boundary of its support t = γ(Ω)− τ (notice that τ could be zero). Since our goal is of local nature we shall need some additional condition which holds, for instance, when f is a bounded function: Let u be the solution of (4.1) and assume (4.18). Let τ ∈ [ , s f ] given by (4.17). Assume data f and Ω, and u be such that, for some δ ∈ ( , γ(Ω) − τ), and where θ > is some constant such that Then (4.24) Notice that due to (4.19) then min s∈(γ(Ω)−τ−δ,min(γ(Ω)− τ ,γ(Ω)) φ Φ − (s) > since min(γ(Ω) − τ , γ(Ω)) < . Moreover, from (4.18) ≤ Then, simplifying the notation K = K δ (Ω) in (4.23), (4.7) and (4.24) we get (4.25) When Ω = R N we recall that the existence of solutions can be proved by well-known methods since the perturbation is sublinear (see e.g. Theorem 4.2 of [15] and Remark 14). Let us construct now a supersolution of (4.9). We de ne the function where η(r) = θr p+ −p with θ satisfying (4.22).
Thus, by the comparison principle, and then which gives the result.

Remark 16.
The above result improves Proposition 5 of [17]. We send the reader to [15] and [17] for many other results concerning solutions with compact support and dead cores, when p < . In particular, it is well known that a suitable balance between the "sizes" of f and the set x ∈ Ω : f (x) = is needed for the occurrence of a free boundary: in some sense the last set must big enough. Such a balance appears here written in terms of the assumptions (4.20) and (4.21). Notice the above results says that if condition (4.20) holds for s = γ(Ω) − τ − δ then we get the decay inequality for any s ∈ (γ(Ω) − τ − δ, γ(Ω) − τ). Finally, notice that if in (4.25) there is an equality, instead an inequality, and if s f > then, necessarily Mu = M f , where

Comparison in mass for problems with a more general non linearity
The results of the previous section can be generalized to a class of elliptic problem with a more general zero order term. Several directions of improvement are possible. We could work with solutions outside the energy space, for instance when f (x)φ(x) ∈ L (Ω), as in the famous paper by Brezis and Strauss [10], but we prefer to continue working with solutions in the energy space and so, to x ideas, we consider in this Section the following generalization where Ω is de ned in (2.6) and f = f , the Gauss rearrangement of f .
The proof of Theorem 17 runs as in the case b(u) = |u| p− u, but as a preliminary step we need that the analogue of Proposition 8 is in force. For reader convenience we detail the following result of existence.

Proposition 18. Suppose that (A ), (A ′ )-(A ′ ) and (B)
hold. If f is nonnegative, then Problem (5.1) has a unique nonnegative weak solution u ∈ H (Ω, γ), i.e. such that c(x) b(u) ∈ L (Ω, γ) and Proof. We give only some details about existence, because the proof of positivity and uniqueness is standards and runs using the monotonicity of b. We introduce the following class of approximated problems: where T k (s) is de ned as in (3.5). Since |T k c(x) b(u k ) |φ(x) ≤ kφ(x) and T k c(x) b(u k ) u k ≥ , the existence of a variational weak solution u k ∈ H (Ω, γ) is well-known (see, e.g. [9]). Taking u k as test function and using Log-Sobolev inequality (2.10) we obtain and for some positive constant C independent of u k . Then the sequence u k is bounded in H (Ω, γ), then there exists a function u ∈ H (Ω, γ) such that (up a subsequence u k u in H (Ω, γ) and u k → u a.e. in Ω hold. In particular By Fatou's Lemma and estimate (5) we get . Moreover for some δ > and every E ⊂ Ω by (5) we get , we get the equintegrability and Vitali's Theorem allow us to conclude that Now we are able to pass to the limit in (5.3) for every ψ ∈ H (Ω, γ)∩L ∞ (Ω) and the result holds.

Remark 19. Theorem 17 also holds if we assume in (B) that b is merely non-decreasing b( ) = and b(u)u >
. The only di culty arises when dealing with b − because now is not necessarily a function but a maximal monotone graph of R and some technicalities are needed (see, e.g., [10], [15] and [35]).

Remark 20.
A di erent extension concerns the case in which we replace f by a general datum F = f − divg with f that satisfy (A ′ ) and g ∈ (L (Ω, γ)) N . To have nonnegative solutions we have to require < F, ψ >≥ for every nonnegative test function.
We can also compare (in the sense of rearrangements) problems with di erent nonlinearities. Just to give an idea, let us consider problem (5.1) when the domain Ω is Hω, the half-space x > ω with ω ∈ R. We take into account two smooth strictly increasing Then, we are going to prove that Hν b(u (x)) dx ≤ Hν b u(x) dx for every ν > ω, (5.6) where u is the rearrangement with respect to Gauss measure of the solution u to problem (5.1) and u is the solution to the following problem We refer to [35] for unweighted case φ(x) ≡ . A more general result, implying conclusion (5.6) can be proved.
To be more precise, let b , b be two continuous non decreasing functions. We say that b is weaker than b , and we write b ≺ b , (5.7) if they have the same domain of de nition, and there exists a contraction ρ : R → R (i.e. such that ρ(a) − ρ(b) ≤ |a − b| for a, b ∈ R) and b = ρ • b (notice that this implies condition (5.5) when they are di erentiable). We are now in position to state a comparison result between the concentration of the solution u ∈ H (Ω) to problem ( . ) with c(x) ≡ (for simplicity) and the solution v ∈ H (Ω , γ) to the following problem where Ω is de ned in (2.  (R N , γ), where A(t) = b(t)t and by asking some additional conditions. We recall that b veri es a ∆ condition, i.e. there exist a constant K and s such that b( s) ≤ K b(s) ∀s > s .
We stress that all estimates in Lemma 3 holds replacing |u| p− u with b(u), when b(ku)ku ≥ |u| p+ for u > u for some k, u > and for some p > and the only crucial step is the weak convergence in the Orlicz space L A loc (R N , γ). Clearly we have to require more on b in order to have that A is an N-function (see [1]). For example b has to be an odd function. Moreover when b veri es a ∆ condition, it is easy to check that A does. Then the Orlicz space L A loc (R N , γ) is re exive and the boundedness of u M L A loc (R N ,γ) allows as to pass to the limit in the sequence of approximate problems.