我们给出了在有界开集中归一化 p-Laplacian 超定问题解的存在性与不存在性结果，其中 p 的范围从 1 到无穷大。更确切地说，我们在边界处考虑非常数 Neumann 条件。理解主要结果所需的定义和命题得到了详细回顾。
英文摘要：
We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail.
We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-Δ_p^Nu=1$ in $Ω$, $u=0$ and $\tfrac{\partial u}{\partialν}=-c\neq 0$ on $\partialΩ$ can only exist on a bounded domain $Ω$ if $Ω$ is a ball.
We consider the $p$-Laplacian equation $-Δ_p u=1$ for $1<p<2$, on a regular bounded domain $Ω\subset\mathbb R^N$, with $N\ge2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\partialΩ$ is constant, then $Ω$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.
We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $Σ$ intersected with the exterior of a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $Σ\capΩ$ must be the intersection of the Wulff shape and $Σ$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.
We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.
The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $Ω\subset \mathbb R^N$, we consider the Riesz-potential $$u(x)=\int_Ω\frac{1}{|x-y|^{N-α}} \,dy$$ for $2\leq α\not =N$. We show that $u=$ constant on the boundary of $Ω$ if and only if $Ω$ is a ball. In the case of $ α=N$, the similar characterization is established for the logarithmic potential. We also prove that such a characterization holds for the logarithmic Riesz potential. This provides a …
We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $Ω$ of $\R^n$ and discuss what kind of implications on the geometry of $Ω$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.
We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neu…
We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in $\mathbb{R}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).
We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth…
In this paper we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{α, \fracα{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.
We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,α}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Radó-type removability theorem.
We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.
We study a tug-of-war game with varying probabilities. In particular, we show that the value of the game is locally asymptotically Hölder continuous. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a solution of the normalized $p(x)$-Laplacian.
We prove that, if $Ω$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $Ω$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.
We give a complete characterization, as "stadium-like domains", of convex subsets $Ω$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $Ω$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet probl…
In this paper, we study an inhomogeneous variant of the normalized $p$-Laplacian evolution which has been recently treated in \cite{BG1}, \cite{Do}, \cite{MPR} and \cite{Ju}. We show that if the initial datum satisfies the pointwise gradient estimate \eqref{e:main1} a.e., then the unique solution to the Cauchy problem \eqref{main5} satisfies the same gradient estimate a.e. for all later times, see \eqref{e:main} below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation \eqref{main5} was first obtained in \cite{CGS}. Such estimate general…
This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations in all the possible cases according to the value of $p$ with respect to the critical exponent $2(1+2/N)$. In the $L^2$-subcritical case, we study a global minimization problem and obtain a ground state solution. While in the $L^2$-critical case, we prove several nonexistence results, extended also in the $L^q$-critical case. At last, we derive a ground state and infinitely many radial solutions in the $L^2$-supercritical case. Compared with the classical Schrödinger equation, the $(2,q)$-Laplaci…
We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.
In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians $$ (-Δ)^{s_1} u +(-Δ)^{s_2} u + λu=|u|^{p-2} u $$ under the constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<s_2<s_1<1$, $2+ \frac {4s_1}{N} \leq p< \infty $ if $N \leq 2s_1$, $2+ \frac {4s_1}{N} \leq p<\frac{2N}{N-2s_1}$ if $N >2s_1$, $λ\in \R$ appearing as Lagrange multiplier is unknown. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^N$, where $\mathcal{F}$ denotes the Fourier…
Normalized groundstates for mixed $(p,2)$-Laplacian equations in $\mathbb R^2$ with exponential critical growth
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $Ω\subset M$ be a bounded domain, with $O \in Ω$, and consider the problem $Δ_p u = -1$ in $Ω$ with $u=0$ on $\partial Ω$, where $Δ_p$ is the $p$-Laplacian of $g$. We prove that if the normal derivative $\partial_νu$ of $u$ along the boundary of $Ω$ is a function of $d$ satisfying suitable conditions, then $Ω$ must be a geodesic ball. In particular, our result applies to open balls of $\mathbb{R}^n$ equipped with a rotationally symmetric metri…
In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -Δ_{p}u+{\vert u\vert}^{p-2}u=f(u)-μu \quad \text{ in } \mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*} where $-Δ_{p}u=div\left( {\vert\nabla u\vert}^{p-2}\nabla u \right)$, $2\leq p<N$, $f\in C(\mathbb{R},\mathbb{R})$, $m>0$, $μ\in\mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfie…
In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-nor…
Given an open bounded subset $Ω$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-Δ_{\infty} u = 1$ in $Ω$, subject to the homogeneous boundary condition $u = 0$ on $\partial Ω$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(Ω)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial Ω$; by using a suitable $P$-function we prove that, if $Ω$ satisfies the same assumptions as above a…
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
