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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16 views

Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...
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0answers
46 views

Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that ...
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1answer
51 views

leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma ...
2
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2answers
135 views

Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies ...
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0answers
151 views
+50

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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0answers
28 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 ...
2
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0answers
41 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
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0answers
64 views

An H2 estimate for Helmholtz equation

How to show the following statement? Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation, $$ -\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u ...
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0answers
30 views

About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines". For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
2
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1answer
252 views

Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
1
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1answer
113 views

Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in ...
1
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1answer
63 views

Why are the tangential derivatives in this diffraction problem zero? [closed]

I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in ...
2
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0answers
121 views

The minimum value of a energy integral

Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0 $ and $${\nabla ...
0
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1answer
79 views

Explicit solution for one-dimensional Gelfand problem

I wonder if the ODE $y''+e^{y}=a$ can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in ...
2
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0answers
32 views

Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold : $ \| e^{\tau \phi} \triangle u \|_{L_{\delta}^2({\mathbb{R^3})}}> C \tau \| e^{\tau \phi} u ...
2
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1answer
42 views

What is the function space $H^1_{m, \sigma}$?

I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order". Theorem 3.1 is the first time in their paper that the function space ...
6
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2answers
188 views

Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details. Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen ...
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0answers
51 views

Bound of solution of pde with a distance function

Hello,i am going to solve the PDE, $\Delta u = -K in \Omega $and u=0 on the boundary ,where K is some poistive constant. Then i have read a paper which stated that u>=distance(x,boundary of ...
2
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0answers
108 views

How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of $$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$ $$\partial_\nu u = \alpha \quad \text{on ...
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0answers
19 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g ...
2
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2answers
199 views

Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function ...
2
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1answer
150 views

Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...
1
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1answer
75 views

Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...
2
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0answers
84 views

Dirichlet-to-Neumann Map is selfadjoint

Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$. For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...
3
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0answers
97 views

Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
2
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0answers
115 views

Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...
0
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0answers
105 views

For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?

Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that ...
7
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1answer
496 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
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0answers
30 views

Elliptic Equation with Wentzell boundary condition

I'm looking for a reference showing how to obtain a priori estimate for solutions to a linear second-order elliptic equation with Wentzell boundary condition in a bounded domain in $H^1$ space. The ...
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0answers
101 views

A priori estimate for diffraction problem for linear elliptic PDEs

I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation. I looked at ...
6
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1answer
161 views

Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it. Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ ...
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0answers
46 views

Regularity result for Neumann problem

I have two questions. On Elliptic regularity for the Neumann problem, the OP asked whether the test function $v$ must be of mean value zero. However, isn't it true that we only need $f$ is of mean ...
3
votes
2answers
136 views

A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy $-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$, where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
3
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0answers
72 views

Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :) Consider the following Laplace boundary value problem (BVP) $$\matrix{ {{\nabla ^2}\Phi ...
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0answers
26 views

Regularity of a flux induced by a potential

Take $\Omega\subset R^n$ with smooth boundary (take a ball for example) a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$ a ...
6
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1answer
209 views

Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert _{W^{s,p}}$, for ...
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1answer
76 views

Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says: Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$. ...
3
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1answer
204 views

stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)? Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} ...
3
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1answer
131 views

dirichlet problem in the heisenberg group

Good morning everybody. I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...
3
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0answers
99 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 ...
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0answers
61 views

How Minimal solution is obtained as limit of approximations

I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this: I can understand the proof, but I don't know why this solution which is obtained as a ...
4
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1answer
93 views

Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted: For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...
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0answers
34 views

Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution to ...
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0answers
96 views

A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
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1answer
68 views

Modified mean value property

Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant. I met this modified mean value property of a solution $u$ of $Lu=0$ as ...
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0answers
139 views

Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
2
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2answers
140 views

Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me. Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...
2
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1answer
221 views

Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and $$ \mathrm{Ricc}(g)=\lambda g, $$ $h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
2
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0answers
63 views

Hypoelliptic pseudodifferential operators and Fredholm equations?

I read here the following: The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable ...
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1answer
116 views

Are solutions of the Beltrami Equations necessarily smooth?

Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The Beltrami Equations are defined as the following system of PDE’s on the domain $ ...