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.

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Applications and motivations of resolvent for elliptic operator

Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is \begin{align} \mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2 \...
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4 votes
0 answers
162 views

Problems arising from the Trudinger's paper in 1968 "Remarks concerning the conformal deformation of riemannian structures on compact manifolds"

I'm reading the paper Remarks concerning the conformal deformation of riemannian structures on compact manifolds by NEIL S. TRUDINGER. I'm stuck with the Theorem 3, which says that let $u$ be a $W_{2}^...
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5 votes
1 answer
112 views

Bernstein's corollary for the case of half space

The early seminal result of Bernstein in 1914 for $n=2$ is the well-known Bernstein theorem: The only entire solutions to the minimal surface equation in $\mathbb R^3$ are the affine functions $$u(x,...
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Relation between the norm of Sobolev space $H^1$ and $L^p$ norm for non-increasing radial functions

I am interested to find $$\sup\|u\|_{p}^{p},$$ when $u$ are non-increasing radial functions on the unit ball $B_1$ of $\mathbb{R}^{n}$ such that $$\|u\|_{H1}^2 < r$$ for some $r > 0$. Since $u$ ...
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4 votes
1 answer
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Smoothness of critical elliptic problem

I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of $$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth ...
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3 votes
0 answers
133 views

$C^1$-regularity of solution of a Dirichlet problem

I've been stuck trying to hunt down a proof/reference for a certain regularity result which I need for my thesis (all the authors I've consulted refer to it as "well-known"). Consider the ...
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1 vote
1 answer
96 views

Does Newton-Leibnitz apply to Sobolev space

For a function $u\in W^{1,p}$, we always use a sequence of $C^1$ function to approach it and derive the consequences. However, can we just claim for a.e. x, y: $$u(x)-u(y)= \int_0^1 Du(y+t(x-y))\cdot (...
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7 votes
2 answers
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Why don't we study hyperbolic equations as elliptic and parabolic equations?

In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ...
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1 vote
1 answer
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About the continuity of the integral on the boundary of a ball

I’m considering a $H^1$ function u on a open domain D. Is the integral: $$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$ continuous with respect to x? I tried to prove that it’s differential by ...
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Fourier spectral methods for an elliptic equation

I would like to study a linear elliptic problem on the torus $\mathbb{T}^n$ (i.e. periodic boundary conditions) which is of the following form: $$ -\Delta u + b^i \partial_i u + c u = f $$ where $b^i$,...
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1 vote
1 answer
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A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used

I'm reading Tawfik - The Yamabe problem: the PDE is $$ \Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1} $$ Theorem (Yamabe). For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\...
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0 answers
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An inequality involving supremum over the boundary

Apologies if this is not fitting for the site, I will remove the post if requested. I am studying elliptic PDE from the book of Chen and Wu, Second Order Elliptic Equations and Systems. I am having ...
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The spectrum of Laplacian operator

Let $ \Omega $ be a bounded domain in $ \mathbb{R}^d $. For $ f\in L^2 $, it is well known that we have a unique solution $ u\in H_0^1(\Omega) $ by using Lax-Milgram theorem for the Dirichlet problem ...
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Poisson Kernel and solution formula for fractional elliptic problem

$$ k (-\Delta)^s u + u = 0, \qquad x \in U, \\ u(x) = f(x), \qquad x \in \mathbb R^n \setminus U, $$ with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
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1 answer
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Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate. I am reading a paper of Brezis and Oswald about existence ...
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4 votes
1 answer
152 views

Conditions for the existence of a solution to a semilinear second-order PDE with a-priori bounds

Consider the general semilinear elliptic second-order PDE $$ u_t-\mathcal L u=f\left(t,x,u,\nabla u\right) $$ where $\mathcal L$ is an elliptic linear operator (like minus the Laplace operator), $t \...
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3 votes
1 answer
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Solution or existence for a second-order semilinear PDE

Consider the following PDE:$$0=u_t+u_{yy}+u_{xx}+(x-y)u_y+y^{-\frac{3}{2}}u^2+1,$$ with $t \in [0,T], $ and a terminal condition $u_T=-1$ for all $x$ and $y.$ The domain for $x$ and $y$ can be bounded ...
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  • 91
4 votes
0 answers
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On the convergence of the spectral decomposition of a harmonic function

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
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2 votes
0 answers
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Reference request – a priori estimate – mixed boundary condition

I am interested in finding references regarding estimates of the form $$ \| D^2 u\|_{L^2(\Omega)} \leq C(\|f\|_{L^2(\Omega)}+\|g\|_{S} )$$ where $\|D^2 u\|_{L^2(\Omega)}^2 = \sum\limits_{i,j \in \{1,2\...
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7 votes
1 answer
481 views

Kernel of the Laplacian + a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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1 vote
0 answers
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Problems arising from a paper on the radial symmetry of the global solution of semilinear PDE $\Delta u+f(u)=0$ in $\Bbb{R}^{n}$

I am reading the paper [1] by Congming Li. I want to talk about the typical case that the author gives as follows ([1], §1, pp. 590-): In this section, we study positive solutions of the following ...
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3 votes
1 answer
165 views

A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture

[This question is looking at the paper Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/...
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3 votes
0 answers
74 views

Differentiability of a weak solution

Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
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  • 703
2 votes
0 answers
296 views

Would you help me to find this expression?

I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate $$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$ Where $\Omega$ is an ...
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1 vote
0 answers
70 views

Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...
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3 votes
0 answers
55 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 ...
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1 vote
0 answers
38 views

Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$

Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
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  • 31
2 votes
1 answer
139 views

The behavior of $ \nabla u $ on the boundary for Poisson equations

Let $ \Omega $ be a bounded domain with smooth boundary. Consider the Poisson equation \begin{eqnarray} -\Delta u&=&f\text{ in }\Omega\\ u&=&0\text{ on }\partial\Omega \end{eqnarray} ...
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0 votes
1 answer
52 views

Non-existence of rapidly decaying solutions of certain elliptic semilinear equations

Consider the equation $$ -\Delta f+mf+\lambda f^p=0$$ on $\mathbb{R}^d$, where $d>2$,$m>0$, $p>1$ is integer, and $\lambda \in \mathbb{R}$. Are there any known results regarding the non-...
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  • 361
2 votes
1 answer
94 views

Spectral analysis for nonlocal elliptic operator

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\...
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  • 267
1 vote
0 answers
36 views

Deriving the general interior elliptic estimate from the compactly supported case

This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The ...
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2 votes
0 answers
71 views

Proving an eigenvalue bound without resorting to Weyl's law

Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
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  • 2,858
3 votes
0 answers
108 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 ...
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  • 361
2 votes
0 answers
39 views

Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator

I would appreciate any answers or even references for the following problem. Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
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  • 2,858
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
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  • 2,858
0 votes
0 answers
34 views

Reference for an argument of existence of a solution of an elliptic PDE problem with lack of compactness

I am reading this paper and the authors stated It is easy to verify that $N_{\lambda}^j, O_{\lambda}^j, \Lambda_{\lambda}^j$ and $\partial \Lambda_{\lambda}^j$ are non-empty sets for $j = 1, \cdots , ...
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  • 425
1 vote
0 answers
59 views

Parabolic/Elliptic equation with nonlinear gradient term

Let $a\in (0,1)$ and $(0,1) \subset \mathbb{R}$, we consider the below equation in $(0,1) \times (0,T)$ $$ \partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$ And $u(x,0)=x^{1/a}$ ...
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  • 178
2 votes
0 answers
71 views

How to approach this semilinear system of PDEs?

This question is cross-posted from Math StackExchange (link). I'm not sure it qualifies as research-level mathematics (although the application is to research) but it has been on MSE for several days, ...
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  • 121
0 votes
0 answers
123 views

Sturm-Liouville result

Suppose $n \ge 2$ an integer and consider finding the first eigenvalue of $$ -\partial_\theta \left( \omega(\theta) \psi'(\theta) \right) = \mu_1 \omega(\theta) \psi(\theta)$$ for $ 0<\theta<\...
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  • 1,313
0 votes
0 answers
24 views

Heat kernel trace estimate with log term

I am studying the heat kernel trace and Weyl law for self-adjoint Hormander operators. For eample, $-\Delta_X:=\sum_{i=1}^2 X_i^*X_i$, where $X=(X_1,X_2)=(\partial_1, x_1\partial_2)$ and $X^*=-{\rm ...
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  • 409
2 votes
0 answers
136 views

Elliptic regularity for a system of PDEs

I am considering a system that can be simplified to the following problem. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$, and consider the following coupled ...
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3 votes
0 answers
46 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 ...
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  • 445
1 vote
0 answers
131 views

Sobolev interpolation inequality for relatively compact subdomains

I was looking at Nicolaescu's Lectures on the Geometry of Manifolds (3rd edition). In Theorem 10.2.29 he presents (without proof) the following inequality: For $m \geq 1, p \geq 1, 0 < r \leq R$ ...
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1 vote
0 answers
61 views

Elliptic systems with two dimensions

Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
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3 votes
1 answer
112 views

$C^{1,\alpha}$ estimate for Newton potential of $L^\infty$ function

Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$ u(x) = \int_\Omega \log |x-y| f(y)\ dy $$ ...
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1 vote
0 answers
95 views

Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation? Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 ...
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  • 215
1 vote
0 answers
52 views

Boundary estimates for elliptic systems

Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
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1 vote
0 answers
41 views

Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
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1 vote
0 answers
49 views

What is the the "method of ascending" in the study of elliptic systems in dimension two?

I have read a paper of Z. Shen [1]. In the paper the author mentioned we can deal with two-dimensional elliptic systems by adding a dummy variable (the method of ascending) and use the results on the ...
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3 votes
0 answers
103 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 ...
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