Optimal Wloc2,2-regularity,Pohozhaevʼs identity,and nonexistence of weak solutions to some quasilinear elliptic equations

We begin by establishing a sharp (optimal) $W_{\mathrm{loc}}^{2,2}$-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, ${\mathrm{\Delta }}_{p}u\stackrel{\mathrm{def}}{=}\mathrm{div}({|\mathrm{?}u|}^{p?2}\mathrm{?}u)$, $1<p<\infty$. We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for $2<p<\infty$) or singular (for $1<p<2$) problem. We apply this regularity result to prove Pohozhaev?s identity for a weak solution $u\in W^{1,p}(\Omega )$ of the elliptic Neumann problem(P)$?{\mathrm{\Delta }}_{p}u+W^{\prime }(u)=f(x)\text{in}\Omega ;?u/?\nu =0\text{on}?\Omega .$ Here, Ω is a bounded domain in ${\mathbb{R}}^N$ whose boundary ?Ω is a $C^2$-manifold, $\nu \equiv \nu (x_0)$ denotes the outer unit normal to ?Ω at $x_0\in ?\Omega$, $x=(x_1,\dots ,x_N)$ is a generic point in Ω, and $f\in L^{\infty }(\Omega )\cap W^{1,1}(\Omega )$. The potential $W:\mathbb{R}\to \mathbb{R}$ is assumed to be of class $C^1$ and of the typical double-well shape of type $W(s)={|1?{|s|}^{\beta }|}^{\alpha }$ for $s\in \mathbb{R}$, where $\alpha ,\beta >1$ are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with $f\equiv 0$ in Ω has no phase transition solution $u\in W^{1,p}(\Omega )$ ($1<p?N$), such that $?1?u?1$ in Ω with $u\equiv ?1$ in ${\Omega }_{?1}$ and $u\equiv 1$ in ${\Omega }_1$, where both ${\Omega }_{?1}$ and ${\Omega }_1$ are some nonempty subdomains of Ω. Such a scenario for u is possible only if $N=1$ and ${\Omega }_{?1}$, ${\Omega }_1$ are finite unions of suitable subintervals of the open interval $\Omega ?{\mathbb{R}}^1$.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

A Hardy type inequality for W02,1(Ω) functions

We consider functions $u\in W_0^{2,1}(\Omega )$, where $\Omega ?{\mathbb{R}}^N$ is a smooth bounded domain. We prove that $\frac{u(x)}{d(x)}\in W_0^{1,1}(\Omega )$ with$6\mathrm{?}\left(\frac{u(x)}{d(x)}\right)6_{L^1(\Omega )}?C6u{6}_{W^{2,1}(\Omega )},$ where d is a smooth positive function which coincides with $\mathrm{dist}(x,?\Omega )$ near ?Ω and C is a constant depending only on d and Ω.     

On sub-harmonic bifurcations

Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family $h_u$ of transformations near $(u_0,x_0)$ when $h_{u_0}(x_0)=x_0$ and $\mathrm{d}{h}_{u_0}(x_0)$ has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Diffusion versus absorption in semilinear parabolic problems

We study the limit, when $k\to \infty$, of the solutions $u=u_k$ of (E) ${?}_{t}u?\mathrm{\Delta }u+h(t)u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $u_k(?,0)=k{\delta }_0$, with $q>1$, $h(t)>0$. If $h(t)={\mathrm{e}}^{?\omega (t)/t}$ where $\omega >0$ satisfies to ${\int }_0^1\sqrt{\omega (t)}{t}^{\mathrm{?1}}\mathrm{d}t<\infty$, the limit function $u_{\infty }$ is a solution of (E) with a single singularity at $(0,0)$, while if $\omega (t)\equiv 1$, $u_{\infty }$ is the maximal solution of (E). We examine similar questions for equations such as ${?}_{t}u?\mathrm{\Delta }{u}^m+h(t)u^q=0$ with $m>1$ and ${?}_{t}u?\mathrm{\Delta }u+h(t){\mathrm{e}}^u=0$. To cite this article: A. Shishkov, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Large solutions to the Monge–Ampère equations with nonlinear gradient terms: Existence and boundary behavior

In this paper, we obtain conditions about the existence and boundary behavior of (strictly) convex solutions to the Monge–Ampère equations with boundary blow-up

$\det ?D^{2}u(x)=b(x)f(u(x))\pm |\mathrm{?}u{|}^q,x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

and

$\det ?D^{2}u(x)=b(x)f(u(x))(1+|\mathrm{?}u{|}^q),x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

where Ω is a strictly convex, bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$, $q\in [0,N]$ (or $q\in [0,N)$), $b\in C^{\infty }(\mathrm{\Omega })$ which is positive in Ω, but may vanish or blow up on the boundary, $f\in C[0,\infty )$, $f(0)=0$, and f is strictly increasing on $[0,\infty )$ (or $f\in C(\mathbb{R})$, $f(s)>0,?s\in \mathbb{R}$, and f is strictly increasing on $\mathbb{R}$).     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

The Lp Robin problem for Laplace equations in Lipschitz and (semi-)convex domains

Let $n\ge 3$ and Ω be a bounded Lipschitz domain in ${\mathbb{R}}^n$. Assume that $p\in (2,\infty )$ and the function $b\in L^{\infty }(?\mathrm{\Omega })$ is non-negative, where ?Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ?Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation $\mathrm{\Delta }u=0$ in Ω with boundary data $?u/?\nu +bu=f\in L^p(?\mathrm{\Omega })$, respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted $L^2(?\mathrm{\Omega })$ space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) $L^p(?\mathrm{\Omega })$ for any given $p\in (1,\infty )$.     

Two remarks on liftings of maps with values into S1

Given a map $u\in L_{\mathrm{loc}}^1(\Omega ,S^1)$ with some regularity: ${|u|}_X=R<\infty$, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u={\mathrm{e}}^{\mathrm{i}\phi }$) with the same regularity and with an optimal control ${|\phi |}_X?g(R)$. Two cases are treated here:(i) ${|?|}_X$ is a $W^{s,p}(0,1)$-seminorm, with $0<s<1<p$ and $sp>1$. We find a lifting φ such that ${|\phi |}_{W^{s,p}(I)}?C(R+R^{1/s})$ and we show that the exponent $1/s$ cannot be improved.(ii) ${|?|}_X$ is the $\mathit{BV}(\Omega )$-seminorm where $\Omega ?{\mathbf{R}}^d$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^1$, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $\phi \in \mathit{BV}(\Omega )$ satisfying ${|\phi |}_{\mathit{BV}}?2R$. To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.     

On nonlinear diffusion problems with strong degeneracy

In this Note, we study the ‘triply’ degenerate problem: $b{(v)}_t?\mathrm{\Delta }g(v)+\mathrm{div}\Phi (v)=f$ on $Q:=(0,T)\times \Omega$, $b(v(0,?))=b(v_0)$ on Ω and $g(v)=g(a)$ ‘on some part of the boundary’ $(0,T)\times ?\Omega$, in the case of continuous nonhomogenous and nonstationary boundary data a. The functions $b,g$ are assumed to be continuous nondecreasing and to verify the normalisation condition $b(0)=g(0)=0$ and the range condition $R(b+g)=\mathbb{R}$. Using monotonicity and penalization methods, we prove existence of a weak entropy solution in the spirit of F. Otto (1996). To cite this article: K. Ammar, C. R. Acad. Sci. Paris, Ser. I 343 (2006).