Half-space Gaussian symmetrization: applications to semilinear elliptic problems

1 Introduction

In this paper we focus our attention on a class of semilinear elliptic Dirichlet problems, whose prototype is

where c₀ > 0, p > 0, Ω is an open subset of ℝ^N not necessary bounded, φ(x)=φN(x):=(2π)−N2exp−|x|22 is the density of standard N−dimensional Gauss measure γ and the datum f belongs to a suitable Zygmund space. A more general diffusion operator is in fact considered in all this paper.

Problem (1.1) is related to Ornstein-Uhlenbeck operator Lu≔ Δ u − x·∇ u and our approach allows us to consider an extra semilinear zero order term c₀∣ u∣^p−1u, Ω=ℝ^N and a weak assumption on the summability of datum. Notice that we can formally write div(∇ u φ(x))=Δ uφ(x)+∇φ(x)·∇ u which justifies the multiplicative role of φ(x) in the equation of (1.1). The idea of “ symmetrizing the operator” −Δu(x)+x·∇ u in order to solve the drift equation

comes back from a pioneering paper [27] by Kolmogorov in 1937 for c₀=0. For some recent survey in this direction see [32]. It is well-known the above diffusion operator with a drift is related to the stochastic Ornstein–Uhlenbeck process with applications in financial mathematics and in physical sciences (a model for the velocity of a massive Brownian particle under the influence of friction). This is sometimes also written in terms of a Langevin ordinary differential equation with noise (see, e.g. [31]).

We remark that if φ ≡ 1 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 > 1).

In the weighted case, when p ⩽ 1 and γ (Ω)<1, Lax-Milgram Theorem guarantees the existence of a solution for problem (1.1) u∈H01(Ω,γ) (the weighted Sobolev space) once we assume f belongs to its dual. For example we can require f∈L2logL−12(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 difficulties that arises when solving (1.1) is due to lack of a Poincaré inequality. As a consequence we have to consider the Banach space H¹(ℝ ^N, γ) equipped with the norm ∣∣ u∣∣_{L²(ℝ^N,γ)}+∣∣∇ u∣∣_{L²(ℝ^N,γ)}. In the linear case, p=1, the existence and uniqueness of a weak solution u ∈ H¹(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 > 1 is different. 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 > 1. We point out that in the superlinear case when γ (Ω)<1 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 > 1 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 infinity on the datum f (and so with less summability than f in L2logL−12(RN,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) modifies 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 (1.1), when γ (Ω)<1, with the solution v to a simpler problem, called the half-space Gaussian symmetrized problem, without zero order term defined in the half-space Ω ^★={(x₁, ..., x_N) ∈ ℝ^N:x₁>ω } with ω such that γ (Ω ^★)=γ (Ω) and with a datum depending only on the first 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 ∈ (0, 1) 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.

2 Preliminaries

Let γ be the N-dimensional Gauss measure on ℝ^N defined by

normalized by γ(ℝ^N)=1. 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 − 1)−rectifiable setEofR^N with respect to the Gauss measureis defined as

where HN−1 denotes the (N − 1)−dimensional Hausdorff measure. The following isoperimetric estimate (see e.g. [11])

holds for all subsets E⊂ℝⁿ, where φ1(s)=2π−12exp−s22 for s∈R and

the 1− 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.

where γ_u(t)=γ({x ∈ Ω:∣ u∣>t}) is the distribution function of u with respect to the Gauss measure. Moreover the rearrangement with respect to Gauss measure of u is defined as

where

and Φ is defined in (2.2).

By definition u^⋆ is a function which depend only on the first 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⊆RN the weighted Lebesgue space L^p (𝛺, 𝛾) with p ≥ 1 is the space of measurable functions u such that

Moreover, as usual, H¹(Ω, γ) states for the weighted Sobolev space of functions u such that u, ∣∇ u∣ ∈ L²(Ω, γ) equipped with the norm ∣∣ u∣∣_L²(Ω,γ)+∣∣∇ u∣∣_L²(Ω,γ). Finally we denote by H01(Ω,γ) the closure of C0∞(Ω) under the norm ∣∣∇ u∣∣_L²(Ω,γ). We remark that a Poincaré inequality holds only when γ (Ω)<1:

for every u∈H01(Ω,γ) , where C_P is a positive constant depending on Ω.

The Sobolev space H01(Ω,γ) is continuously embedded in the Zygmund space L2(logL)12(Ω,γ) (see [26], [24], [23] and references therein). We recall that given 1 ⩽ p<∞ and −∞<α<+∞, a measurable function u belongs to the Zygmund space L^p(logL)^α(Ω, γ) if

Then, it is well-known that there exists a constant C_S depending on Ω such that

for all u∈H01(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 refinement of the usual Lebesgue spaces. Indeed by definition (2.9) the space L^p(logL)⁰(Ω, γ)=L^p(Ω, γ). For the definition 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 H¹(ℝ^N, γ).

3 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 definition of weak solution. We refer to [8,19] for unweighted case φ(x) ≡ 1.

We introduce the natural energy space for the linear problem (for the case p=1)

where H_ω is defined in (2.3). We stress that for a given function f such that f∈L2(logL)−12Hω,γ for any H_ω the integrals

are well-defined for any ψ ∈ V (ℝ^N) (even if f is not necessarily in the dual space of H¹(ℝ^N, γ)).

In this section we will assume the following structural conditions:

1. p > 1

2. aijφ∈L∞(Hω)∀ω∈R and ∑i,j=1Naij(x)ξiξj≥αφ(x)|ξ|2 for a.e. x∈RN , ∀ ξ ∈ ℝ^Nwith α>0

3. c ∈ L¹(H_ω, γ) and c(x) ⩾ c₀ > 0 for x ∈ H_ω for any ω∈R

4. f∈L2(logL)−12(Hω,γ) for any ω∈R .

proof. Indeed by (A2) and (A3) we get

and

Using (3.6), (3.9) and (3.10) we get

Since p > 1, by Young inequality we get that

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 k₀, k₁, k₂, k₃, k₄. As before we get

for some positive constant δ′ that can be chosen later. Taking θω′(x1)=O|ω−x1|m−1 with m > 0, then

for some positive constant k₅ and the last integral is finite if m>p+1p−1 . Moreover

for some positive constant k₆ and the last integral is finite if m > 1. Choosing ε, ε′, δ and δ′ small enough and using (3.16), (3.17), (3.15), (3.14), (3.13), (3.12) in (3.11) we get

for some positive constant k₇, k₈. 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 H¹(H_ω+ω0,γ) and it follows that there exists u such that (up a subsequence) u_M → u weakly in H¹(H_ω, γ) for any ω∈R , weakly in L^p+1(H_ω, γ) for any ω∈R , strongly in L^q(H_ω, γ) for any ω∈R with q < p+1 and a.e. in H_ω for any ω∈R . Using these convergences and the monotonicity of function G(s)=∣s∣^p−1s we can pass to the limit in (3.3) and we conclude.

Step 2. Uniqueness. Let u₁ and u₂ be two different weak solutions to Problem (3.1) such that c∣u₁∣^p+1, c∣u₂∣^p+1 ∈ L¹(H_ω, γ) for any ω∈R . We stress that they satisfy (3.2) with ψ=u₁Θ_ω and ψ=u₂Θ_ω for any ω∈R . Let v=u₁−u₂. Since v ∈ H¹(H_ω, γ)\normalcolor∩ L^p+1(H_ω, γ) for any ω∈R we get

where

For p > 1 we have Γ(x)=c₀ a.e. in ℝ^N. Then Lemma 3 can be applied obtaining

and

for some positive constant K₁ . Lebesgue’s dominated convergence theorem allows

Putting ω goes to −∞ we get ∇ v=0 a.e. on ℝ^N by (3.19) grad v and then v=0 a.e. on ℝ^N using (3.20). This is a contradiction and the uniqueness result follows.

4 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 defined in an half-space having data depending only on one variable.

We need starting with the case γ (Ω)<1 and we consider the following class of Dirichlet problems

The structural assumptions (instead of (A1)−(A4)) are now the following:

(A0) Ω is an open subset of ℝ^N (N=2) such that γ (Ω)<1,

(A1′) p > 0,

(A2′) aijφ∈L∞(Ω) and ∑i,j=1N a_ij(x)ξ_iξ_j ⩾ αφ(x)∣ξ∣² for a.e. x ∈ Ω, ∀ξ ∈ ℝ^N with α>0,

(A3′) c ∈ L¹(Ω, γ), c(x) ⩾ c₀ > 0,

(A4′)f ∈ L²logL^−1/2(Ω, γ).

We recall that under assumption (A2′) a Poincaré inequality holds and that f ∈ L²logL^−1/2(Ω, γ) can be identified with an element in the dual space of H01(Ω,γ) (Ω, γ) (see [4]). Then, under our assumptions, all terms in the corresponding notion of weak formulation are well-defined using (2.10) and (2.8). Sinceγ (Ω)<1, 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 first result of this section shows a suitable integral comparison between the solution u to problem (3.1) and the solution v to the following symmetrized problem

where Ω^★ is the half-space defined in (2.6) and f˜ is such that f˜=f˜★ , the rearrangement with respect to Gauss measure of f˜ defined in (2.5).

First of all, we prove that the solution to (4.2) coincides with its half-space Gaussian rearrangement.

Now we are in position to prove the following comparison result.