A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.
In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].
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.
Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $Ω\subset \mathbb{R}^2$ a $C^{1,α}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$ \left\{\begin{array} {ll} Δu + f(u) = 0 & \mbox{in }\; Ω \\ u= 0\, \, \, , \, \, \, \frac{\partial u}{\partial \vecν}=1 &\mbox{on }\; \partial Ω\end{array}\right. $$ we prove that $Ω$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found…
We study the existence of non-trivial unbounded domains of $Ω\subset \mathbb{R}^2$ where the equation \begin{align} - λu_{xx} -u_{tt} &= u \qquad \text{in $Ω$,}\nonumber u &=0 \qquad \text{on $\partial Ω$,}\nonumber \end{align} is solvable subject to the conditions \begin{align} \frac{\partial u}{\partial η} =-1\quad \text{on $\partial Ω^+$} \quad \textrm{and}\quad \frac{\partial u}{\partial η} =+1\quad \text{on $\partial Ω^-$.} \end{align} For every integer $m\geq 0$, we prove the existence of a family of unbounded domains $Ω\subset \mathbb{R}^2$ indexed by $0 \leqslant\ell\leqslant 2m$, wher…
In this paper, we prove the existence of nontrivial unbounded domains $Ω\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} Δu +f(u)=0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=\mbox{constant} &\mbox{on $\partialΩ$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +\infty) \to \mathbb{R}$. Roughly speaking, we only ask that the D…
We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.
In a bounded domain $Ω$, we consider a positive solution of the problem $Δu+f(u)=0$ in $Ω$, $u=0$ on $\partialΩ$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $Ω$ (for instance, if $Ω$ is convex), we show that $\partialΩ$ is contained in a spherical annulus of radii $r_i<r_e$, where $r_e-r_i\leq C\,[u_ν]_{\partialΩ}^α$ for some constants $C>0$ and $α\in (0,1]$. Here, $[u_ν]_{\partialΩ}$ is the Lipschitz seminorm on $\partialΩ$ of the normal derivative of $u$. This result improves to Hölder stability the logarithmic estimate obtain…
In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze-Karcher-Ros type inequality for embedded hypersurfaces with free boundary lying on a horosphere …
We study Serrin's overdetermined boundary value problem \begin{equation*} -Δ_{S^N}\, u=1 \quad \text{ in $Ω$},\qquad u=0, \; \partial_ηu=\textrm{const} \quad \text{on $\partial Ω$} \end{equation*} in subdomains $Ω$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$, where $Δ_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A subdomain $Ω$ of $S^N$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result …
In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $ω$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for non-quasianalytic weight functions $ω$. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragmén-Lindelöf principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.
We consider the solution of the torsion problem $-Δu=1$ in $Ω$ and $u=0$ on $\partial Ω$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $\partial Ω$, then $Ω$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$ r_e-r_i\leq C_t\,\Bigl(\max_{Γ_t} u-\min_{Γ_t} u\Bigr) $$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partialΩ$ and $Γ_…
We study the existence of nontrivial unbounded domains $Ω$ in $\mathbb{R}^N$ such that the overdetermined problem $$ -Δu = 1 \quad \text{in $Ω$}, \qquad u=0, \quad \partial_νu=\textrm{const} \qquad \text{on $\partial Ω$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that the…
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the interior ball condition and $c>0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on …
This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constr…
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 show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.
Bifurcation of overdetermined capillary problems in a strip domain
Universidad de Granada（格拉纳达大学）；IMAG, Departamento de Análisis matemático, Universidad de Granada（IMAG，数学分析系，格拉纳达大学）；Aix Marseille Université - CNRS, Centrale Marseille - I2M（艾克斯马赛大学-法国国家科学研究中心，中央马赛学院-数学与理论物理研究所）
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work with T.P. Branson, M.G. Eastwood, and A.R. Gover. First a simple special case is discussed, then the (Riemannian) procedure is described in general. The prolongation procedure was derived from a simplification of the construction of Bernstein-Gelfand-Gelfand (BGG) sequences of i…
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…
Overdetermined problems for the rotationally invariant Poisson equation in model manifolds
Università degli Studi di Cagliari（卡利亚里大学）；College of Staten Island, City University of New York（纽约市立学院）；Universidad de Granada（格拉纳达大学）；Aix Marseille Université - CNRS, Centrale Marseille - I2M（艾克斯马赛大学-法国国家科学研究中心，马赛中央理工学院-数学研究所）
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie algebras of vector fields and their algebraic-geometric representations are involved in solving overdetermined of PDE and getting integral representation of stochastic differential equations (SDE). It is addressing to all scientists using PDE in treating mathematical methods.
Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type, and in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $Ω$ is not simply connected, or if $u$ is $C^{\infty}$ but not real analytic.
We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single linear Partial Differential Equation (PDE) with potentials expressed through the field of the nonlinear PDE. The absence of linear overdetermined system associated with nonlinear PDE creates an obstacle to obtain evolution of the spectral data (or dressing functions): evolution is defined by nonlinear IDE (or PDE in particular case). As an example, we consider g…
Linear first order systems of partial differential equations of the form $\nabla f = M\nabla g,$ where $M$ is a constant matrix, are studied on vector spaces over the fields of real and complex numbers, respectively. The Cauchy--Riemann equations belong to this class. We introduce a bilinear $*$-multiplication on the solution space, which plays the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equations $\nabla f = M\nabla g$ constitute only a simple special case of a much larger class of systems of parti…
The equation arising from Prandtl boundary layer theory is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since a may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kruzkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, the stability can be proved ev…
We show uniqueness for overdetermined elliptic problems defined on topological disks $Ω$ with $C^2$ boundary, i.e., positive solutions $u$ to $Δu + f(u)=0$ in $Ω\subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte $ along $\partial Ω$, $\vecη$ the unit outward normal along $\partialΩ$ under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. When $(M^2,g)$ is the standard sphere $\mathbb S^2$ and $f$ is a $C^1$ function so that $f(x)>0$ and $f(x)\ge x \,…
We investigate the application of windowed Fourier frames (WFFs) to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given by the symbol of the PDO evaluated at the wave number and central position of the windowed plane wave. This can be exploited in a preconditioning method for use in iterative inversion. For domains with periodic boundary conditions we find that the condition number with the preconditioning bec…
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as detour complexes. This elaborates on a lecture given at the IMA Summer Programme ``Symmetries and overdetermined systems of partial differential equations''.
It is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain, with Dirichlet boundary conditions, are unique up to multiplication by a positive constant.
We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $$φ(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \in \partial Ω$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \to + \infty$ of Serrin's symmetry problem for the $p$-Laplacian),…
