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

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32 views

Gaussian curvature of conformal transformations

Let $g$ be a smooth metric and $g'=e^{v}g$, where $v$ is also a smooth function. Then it is well-known that $(*) -\Delta_g u +2k_g=2 k_{g'}e^{v}$, where $k_g$ and $k_{g'}$ are the Gaussian ...
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1answer
19 views

Generation of strictly contraction Semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
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0answers
33 views

Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
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2answers
126 views
+50

Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Where can I find a (readable and self-contained) proof of the following result? Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
2
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0answers
93 views

DeGiorgi oscillation lemma

Where can I find a proof of the following result? Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$ where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(...
3
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1answer
108 views

Global regularity for Neumann problem

Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
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0answers
58 views

Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary

In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669 Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
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0answers
37 views

Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE $$ (\Delta-k^{2})u=-g $$ on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
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0answers
187 views

Spectral Gap of Elliptical Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of elliptical operator $ \nabla \cdot(a(x)\nabla)$ defined on $D$, can be controlled? The boundary condition is that the solution at ...
5
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3answers
191 views

about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
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0answers
67 views

Green functions for circular sectors

I would like to solve the Dirichlet boundary value problem $$ (\Delta-k^2)u=0 \ \ \ \text{in $\Omega$} \\ u=f \ \ \ \text{on $\partial \Omega$} $$ where $\Omega$ is an infinite circular sector of ...
4
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2answers
214 views

Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs, where $P_k$ is given by \begin{equation} \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
5
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0answers
109 views

Extension of elliptic complex to an exact sequence

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator. Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
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0answers
24 views

Eigenfunction of Distorted Laplacian on Smooth compact domain

Setup Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...
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0answers
47 views

Solving Oblique Boundary value problem using Neumann Boundary Condition

\begin{align*} &\frac{1}{2}a_{ij}(x)\frac{\partial^{2}u }{\partial x_{i}\partial x_{j}}+ b_{i}(x)\frac{\partial u}{\partial x_{i}}+ c(x)u=\lambda u ~~~in ~~\Omega\\ &\frac{\partial u}{\partial ...
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0answers
65 views

Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...
6
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1answer
197 views

is signed distance function real analytic for real analytic domains

If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by \begin{equation} f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\...
5
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1answer
102 views

Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell: Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form? Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
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0answers
50 views

Convergence in integral

I tried to prove that: (f1) $f(0)=0$, $f\in C^1(0,\infty)$, $f'(s)>0$ for $s>0$, and there is a $p\in(2, 2N/(N-4))$ such that $\lim_{s\to \infty}f(s)/s^{p-2}<\infty$, (f2) $F(s)=\int_0^sf(t)...
2
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1answer
111 views

Evans-Krylov theorem

Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
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0answers
46 views

Degeneracy of critical points of solutions to second order elliptic PDEs

Problem: Suppose that $Lu = f(x)$ on a strictly convex domain $\Omega\subset\mathbb{R}^2$ and $u=0$ on $\partial\Omega$ where $L$ is elliptic and second order. I am trying to determine when the ...
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17 views

Are solutions to the second order elliptic PDE $Lu=f(u,\nabla u, x)$ locally $L$-harmonic homogeneous polynomials?

These two papers by Caffarelli and Friedman [1], [2] show that a solution $u$ to the elliptic PDE $$ \Delta u = f(u,\,\nabla u,\, x)\,,\quad \text{ on } \Omega\subset\mathbb{R}^n $$ is locally a ...
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0answers
66 views

Anderson Localization and Homogenization theory

I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here. The question is mostly related to homogenization theory in mathematical physics. $\textbf{...
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1answer
103 views

Existence of unique critical points to second order elliptic PDEs

Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1) $$ \begin{cases} Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
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0answers
37 views

Elliptic Dirichlet BVP's for regions with multiple boundary components

Apologies for the vague title, it was getting rather long so I decided to just explain more in the body of the text. I am curious about the state of understanding for existence and uniqueness of ...
1
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1answer
114 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$. Explicitly, let $\M$ be a smooth oriented Riemannian manifold. ...
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0answers
34 views

First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
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0answers
103 views

Analytical solve for this 4th order Linear PDE

I want to Solve analyticaly or numericaly a biharmonic equation with homogeneous boundary conditions homogeneous in rectangular domain [0,a]*[0,b]? I considered its solution using Fourier analysis (...
3
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1answer
147 views

Does the space of harmonic forms change continuously with the metric?

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...
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0answers
28 views

Change of variable for the Stokes equations

I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741). Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...
2
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1answer
156 views

On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite ...
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0answers
44 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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2answers
234 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
6
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1answer
211 views

Reference Question: Boundary Value Problem for Dirac Operator on Manifold with a non-smooth boundary

I am trying to find references for the following boundary value problem: Assume that $\Omega$ is a compact 3-dim spin manifold with Dirac operator $D$ such that the boundary consists of two smooth ...
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78 views

Counter-examples to the higher dimensional statement of the half-space theorem

The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$. The higher dimensional ...
2
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1answer
153 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
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1answer
70 views

PDE satisfied by projection of a function onto a subspace

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}...
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0answers
44 views

Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$. In the ...
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2answers
70 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
0
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1answer
67 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
3
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0answers
83 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 ...
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1answer
89 views

Comparison principle for viscosity solution

I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen. https://projecteuclid.org/euclid.jdg/1090349447 I ...
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0answers
37 views

Value at $(0,0)$ of the solution to a 2D linear elliptic PDE

The setup is the following. Let $D=(-\frac{1}{2} , \frac{1}{2})^2 \subset \mathbb{R}^2$, $\partial D$ be its boundary. Let $a, b\in \mathbb{R}$ and $f$ be continuous on $\partial D$. Also we assume ...
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0answers
29 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
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0answers
48 views

Conformal factors and light rays

Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $...
1
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1answer
166 views

Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
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2answers
126 views

Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
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0answers
54 views

Continuity of solution to 2nd Order PDE w.r.t. the coefficients

I am considering the following 2nd order PDE : On a domain $R$, for some $r > 0$, \begin{equation*} \frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...
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0answers
43 views

Regularity for Elliptic equation with mixed boundary condition in one dimensional domain [0,1]

I am stuck at establishing regularity for elliptic equations on the one dimensional domain $\Omega=[0,1]$. The problem is $Lu=f$, in $\Omega$, and $u(0)=0, u_x(1)=0.$ In the page 317, Evans' book, ...
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3answers
491 views

Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...