Questions tagged [elliptic-pde]

Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

518 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
3 votes
0 answers
63 views

Existence of ground state solutions for the critical exponent

I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$. The authors show that the above equation has a unique positive ...
Student's user avatar
  • 653
3 votes
0 answers
151 views

A version of the Nash-Moser inverse function for unbounded domains?

Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
S.Z.'s user avatar
  • 463
3 votes
0 answers
63 views

Eigenvalues of an elliptic operator on shrinking domains

This was probably done somewhere 100 times, but I can't find a reference. Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
Ivan's user avatar
  • 445
3 votes
0 answers
356 views

Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial ...
student's user avatar
  • 1,320
3 votes
0 answers
191 views

The regularity of solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary

While doing my research, I encountered the following problem as: is there any regularity result for solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary? For ...
yimin's user avatar
  • 31
3 votes
0 answers
95 views

A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$

Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball. Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta ...
Laithy's user avatar
  • 865
3 votes
0 answers
83 views

Minimal normal graph

Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
Eduardo Longa's user avatar
3 votes
0 answers
131 views

Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball

Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption: (H1): There is a dilation structure $$\delta_{t}:\mathbb{R}^n\to \...
Houa's user avatar
  • 561
3 votes
0 answers
196 views

integration by parts on a Lipschitz domain as $\epsilon\to 0$

For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that $$ \lim\limits_{\epsilon\to ...
leo monsaingeon's user avatar
3 votes
0 answers
270 views

Integral representation of solution of an elliptic PDE in divergence form

Suppose we have a second order elliptic differential operator $$ L(v) = -\text{div}(A(x) \nabla v) $$ $A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
Harish's user avatar
  • 251
3 votes
0 answers
185 views

Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator. Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
sharpe's user avatar
  • 701
3 votes
0 answers
76 views

Reference requests: $W_{p}^1$-estimate for $(\triangle -\lambda)$ on Lipschitz domains

Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\...
Will Kwon's user avatar
  • 323
3 votes
0 answers
64 views

Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
Ali's user avatar
  • 4,077
3 votes
0 answers
214 views

Does the minimal surface system in the plane have the weak unique continuation property?

Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
SBK's user avatar
  • 1,141
3 votes
0 answers
183 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
Tao's user avatar
  • 419
3 votes
0 answers
55 views

Harnack inequality for $F(D^{2}u, Du,x)=f$

I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$, ...
Cézar Bezerr's user avatar
3 votes
0 answers
105 views

Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation: $(\omega_0 +i \partial \bar \partial \varphi)^...
Chris's user avatar
  • 31
3 votes
0 answers
119 views

Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
Jason's user avatar
  • 59
3 votes
0 answers
94 views

About p-laplacian and variations

Let $\Omega \subset \mathbb{R^{n}}$ be a domain (open and connected set), for $p\geq 2$, the $p$-laplacian is defined by: $\Delta_p u= \operatorname{div} (|\nabla u|^{p-2} \nabla u)$, in non-...
Cézar Bezerr's user avatar
3 votes
0 answers
95 views

Partial regularity of harmonic maps into spheres

Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
SMS's user avatar
  • 1,293
3 votes
0 answers
191 views

Singular integral operators and PDEs

What is the relation between the notion of singular integral operators and partial differential equations? I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
XIII's user avatar
  • 707
3 votes
0 answers
86 views

Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain

Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
Khoa Le's user avatar
  • 31
3 votes
0 answers
67 views

Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as $$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$ where $X_0,X_1,...,X_k$ are ...
S.Surace's user avatar
  • 1,675
3 votes
0 answers
127 views

Unique continuation from the boundary for inhomogeneous elliptic pde

Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
SMS's user avatar
  • 1,293
3 votes
0 answers
162 views

Asymptotic behaviour of principal eigenfunctions and large deviations

Dear Math Overflowers, I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
leo monsaingeon's user avatar
3 votes
0 answers
105 views

metric with curvature bounded in $L^2$

My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates ...
Paul's user avatar
  • 914
3 votes
0 answers
112 views

Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
geometricK's user avatar
  • 1,851
3 votes
0 answers
79 views

Generalized viscosity sub(super)solution and it's convolution

Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant. Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...
Pan's user avatar
  • 167
3 votes
0 answers
84 views

Estimate a function given an estimate of its Laplacian

Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions: \begin{eqnarray*} \int_B |f_\lambda(x)|^2dx\leq 1,...
asv's user avatar
  • 21.1k
3 votes
0 answers
73 views

Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry

The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...
Dickson's user avatar
  • 31
3 votes
0 answers
120 views

Partial regularity for transmission problem in corner domains

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
PeteAgor's user avatar
  • 133
3 votes
0 answers
182 views

Families of unbounded operators

Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
Joey's user avatar
  • 331
3 votes
0 answers
72 views

Elliptic operator applied to the distance function

Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator: $$ \Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega) $$ where $A(x)$ is a ...
Onil90's user avatar
  • 823
3 votes
0 answers
145 views

When a PDE add a Laplacian term

I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
qie wen's user avatar
  • 39
3 votes
0 answers
260 views

Principal eigenvalue of Laplacian under volume preserving mean curvature flow

Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some ...
Enhao Lan's user avatar
  • 165
3 votes
0 answers
141 views

Prove the positivity of the subelliptic operator heat kernel

Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a ...
pxchg1200's user avatar
  • 265
3 votes
0 answers
553 views

2D laplace equation with Robin boundary condition (Green function)

Let's say that I know a fundamental solution for the Laplace equation in the whole plane: $$\nabla^2u=\delta\quad \text{in the sense of distributions,}$$ and I need a solution for the laplace equation ...
Manuel Pena's user avatar
3 votes
0 answers
179 views

Characterization of minimizer of convex functional

I need to check whether the following characterization of the minimizer of a convex functional is valid. Let $X$ be a reflexive Banach space (think $W^{1,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ ...
D G's user avatar
  • 201
3 votes
0 answers
124 views

Wave equation that becomes elliptic on a bounded domain (sign-changing coefficient)

I'm looking for results on this kind of problems: $$ \partial_{tt}^2 u - \partial_x(a(x) \partial_x) = f,$$ $$u(t=0) = u_0, \quad \partial_t u(t=0) = u_1,$$ where $a$ changes sign: $a(x)= -c^2 < 0$ ...
Héhéhé's user avatar
  • 595
3 votes
0 answers
133 views

Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE. I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of $$\inf_{a \in [-1,1]} \{...
kenneth's user avatar
  • 1,369
3 votes
0 answers
107 views

A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation: $$ -\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$ Here $a_i:...
user avatar
3 votes
0 answers
73 views

On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$: \...
5th decile's user avatar
  • 1,461
3 votes
0 answers
244 views

Interior regularity for elliptic operators with non smooth coefficients

I need a pretty standard interior regularity result for a second order elliptic operator of the form $$ -\nabla^b \cdot (A(x) \nabla^b v)+c v=f, \qquad \nabla^b=\nabla+ib(x) $$ where $A(x)$ is a ...
Piero D'Ancona's user avatar
3 votes
0 answers
134 views

Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations: $(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$ $(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$ for $...
Coffee's user avatar
  • 601
3 votes
0 answers
127 views

Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & \mathrm{in}‎\hspace{...
Hheepp's user avatar
  • 361
3 votes
0 answers
161 views

probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions: Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...
megaproba's user avatar
  • 353
3 votes
0 answers
324 views

Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain

Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
jmk's user avatar
  • 315
3 votes
0 answers
118 views

A question about fractional integrals

I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...
TravelingBoy's user avatar
3 votes
0 answers
263 views

Result like Brezis-Kato Lemma for Biharmonic equation.

Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\ $\Delta^2 u = a(x)u$ and $a(x)\in L^{\frac{n}{4}}(\Omega)$, where $\Omega$ is bounded. If $...
Abhishek's user avatar
3 votes
0 answers
379 views

problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution? $$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 (\...
Regina's user avatar
  • 41

1 2
3
4 5
11