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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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12 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 ...
2
votes
2answers
68 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
votes
1answer
148 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 ...
6
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0answers
67 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
125 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 $$\...
3
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0answers
41 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 $D$}...
2
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0answers
41 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
63 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 ...
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1answer
33 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 ...
1
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1answer
84 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
24 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 ...
1
vote
0answers
47 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
154 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
113 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
51 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
37 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, ...
10
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3answers
460 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{...
1
vote
1answer
45 views

Reference request: Existence/regularity for viscous Hamilton-Jacobi equations

A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as: \begin{equation*} u - \epsilon \Delta u + H(Du) = f(x) \end{equation*} or \begin{equation*} u_{t} -...
3
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0answers
62 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 $T$. ...
4
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0answers
93 views

Davies' definition of elliptic operators in “Heat Kernels and Spectral Theory”

I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
6
votes
1answer
61 views

Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds

In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
2
votes
1answer
129 views

Local existence of non-trivial solutions to first order linear elliptic system of pde

This question came up when I was trying to find out the details about the existence of isothermal coordinates for surfaces. Given a surface in $\mathbb{R}^3$, at least $C^2$ for simplicity, at any ...
4
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0answers
111 views

Is this a pseudodifferential operator?

Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator $$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$ a ...
2
votes
0answers
209 views

Siu's arguments on Calabi-Yau theorem?

In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...
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0answers
27 views

Higher regularity of weak solution of Laplace equation with Robin condition

From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP \begin{align} -\Delta u &= 0 & \text{in}\ \Omega\\ -\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\ \end{align} where $a&...
0
votes
1answer
66 views

Regularity of Laplace equation with Dirichlet data on a part of the boundary

From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system \begin{align} -\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\ u &= g &\text{on}\ \...
0
votes
1answer
52 views

$H^2$ regularity for Laplace equation with Robin-Robin boundary condition

From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP \begin{align} -\Delta u &= 0 & \text{in}\ \Omega\\ -\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\ \end{align} where $a&...
2
votes
1answer
73 views

Higher regularity of solutions for Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\...
5
votes
1answer
165 views

Hodge Laplacian in local coordinates

On a Riemannian Manifold $M^n$, the Hodge Laplacian is defined on k-forms by $\Delta\omega=\operatorname{d}\operatorname{d}^*\omega+\operatorname{d}^*\operatorname{d}\omega$. For 0-forms, e.i. smooth ...
1
vote
1answer
98 views

Inverse of holomorphic elliptic differential operator

Consider the Beltrami-Laplacian $\Delta$ on $\mathbb{S}^n$ with standard metric. One can define a family of operators $A(z):H^1(\mathbb{S}^n)\to H^1(\mathbb{S}^n)$ as the following $$A(z)=\Delta+z$$ ...
2
votes
1answer
126 views

Barry Simon's decay of eigenfunctions for pseudo differential operators

In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result. Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the ...
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0answers
44 views

On the solution of Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system $$ \begin{...
4
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0answers
51 views

Sharp assumption on domain for elliptic boundary regularity

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded Lipschitz domain. Let $f\in H^{s}(\Omega)$, for $s\geq -1$. Then Lax-Milgram guarantees a unique solution to the Poisson problem $$\begin{cases}\Delta u=...
3
votes
1answer
187 views

Regularity of solutions to $-\Delta u = \operatorname{div} F$, $F\in L^1$

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. What are the regularity results for solutions to $$ -\Delta u= \operatorname{div} F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$...
4
votes
0answers
108 views

global estimate for biharmonic function

My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...
4
votes
1answer
103 views

Is the maximal function bounded on the Besov space?

The Hardy-Littlewood maximal function of a function $f$ is defined by $$ M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy, $$ where $|B_r|$ denotes the Lebesgue measure of the ball $...
2
votes
0answers
71 views

Ergodic type ODE problem

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
3
votes
0answers
53 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 > ...
1
vote
1answer
86 views

schauder regularity heat equation

Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$. It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
4
votes
1answer
145 views

Convolution of viscosity solutions and subharmonic functions

Suppose that $u : \mathbb{C}^n \rightarrow \mathbb{R}$ is continuous. We say that $u$ is a viscosity subsolution (resp. viscosity supersolution) for the Laplace's equation if for all $\varphi \in C^2$ ...
2
votes
1answer
100 views

Exponential Decay on nolinear Schrodinger type equation with negative potential

Given a unbounded domain $\Omega$(a region or an open manifold) with $\dim \geq3$, consider the equation $$A u+Vu+|u|^ku=0,$$ here $A=\sum_{ij}\partial_ia^{ij}(x)\partial_j$ is an elliptic operator ...
3
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0answers
73 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,...
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vote
0answers
61 views

Coercivity of $\int (\Delta u + u)^2$ on a subspace of $H^2$?

Let $\Omega = [0,L] \times [0,2\pi]$ and split its boundary into $\Gamma_d = \{0,L\} \times [0,2\pi]$, $\Gamma^1_p = [0,L] \times \{0\}$, $\Gamma^2_p = [0,L] \times\{2\pi\}$. Consider the following ...
3
votes
1answer
219 views

Existence en regularity of elliptic PDE with mixed boundary

Let $\Omega=\mathbb{D}\cap\{ (x,y)\, \vert\, y>0\}$, $I=(-1,1)\times \{0\}$ and $A=\partial\Omega\setminus I$. Let $Q\in L^1(\Omega)$, and $R\in C^\infty_{loc}(I)$. I am looking to the following ...
2
votes
1answer
51 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
1
vote
1answer
68 views

Vorticity equation for generalized Naiver Stokes equations

In $3D$ or $2D$, I can get the vorticity equation for the incompressible NSE; however, what's the vorticity equation for the generalized NSE? Does the fractional laplacian commute with the curl?
2
votes
1answer
107 views

Solving classical parabolic equation by using Littlewood-Paley theory

Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the ...
0
votes
0answers
56 views

Estimate of $\|\nabla u\|_{L^{\infty}(\mathbb R^2)}$ for incompressible Navier-Stokes equations

Consider the incompressible Navier-Stokes equations on $\mathbb R^2:~u_t - \Delta u + u\cdot\nabla u + \nabla p=f$. I want to estimate $\|\nabla u\|_{L^{\infty}(\mathbb R^2)}$. Is there a way to ...