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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25 views

Regularity of Laplace equation on non-convex polyhedral domain

This might be a known problem, but I could not find a precise answer. I have the following Laplace equation \begin{equation} \begin{cases} -\Delta u = f & x \in \Omega;\\ \quad\: u = g & x \in ...
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1answer
63 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
3
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1answer
83 views

Reaction-diffusion systems treated as dynamical systems

I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems. I have the book of Alain Haraux – Systèmes dynamiques dissipatifs et ...
2
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1answer
77 views

Reference request on Pucci extremal operators

While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by: $$M_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{...
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0answers
97 views

Determine the location of the boundary where the heat changes fastest

I am motivated by the following question: Given a uniform heat source in a convex domain, and suppose that the outside temperature is equal to $0$, can we determine where the long-time temperature ...
2
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1answer
156 views

Reference request for semilinear PDEs in dimension 2

I am interested in the study of the (semi-linear, I suppose) equation $$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\ u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$ ...
3
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0answers
134 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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
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32 views

On the short distance behavior of the Green functions of powers of the Laplacian

Let $M$ be a closed Riemannian manifold and let $\Delta=dd^{*}$ be the (positive) Laplacian on $M$. Given $\lambda>0$ and a positive integer $s$, set $G_{\lambda,s}=(\Delta^s+\lambda)^{-1}$. ...
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0answers
44 views

Fractional Laplacian with non-homogeneous boundary condition $u = c \text{ on } \mathbb R^N \setminus \Omega$

We consider $$ \begin{cases} (-\Delta)^s u = 0 &\text{in } \Omega \\ u = c & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ where $(-\Delta)^s$ is the fractional Laplacian operator and ...
5
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1answer
76 views

Does a suitable famlly of eigenvectors of non self-adjoint operators, sufficiently close to an adjoint one, form a basis?

It is very well known that if $A\in L^\infty(B_1;\mathbb R ^{d\times d})$ is a positive definite symmetric matrix, the eigenvalue of the self adjoint operator $H^2(B_1)\cap H^1_0(B_1)\to L^2(B_1)$ $$T:...
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40 views

Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?

Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
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0answers
28 views

Strong convergence from elliptic equation

Consider the following equation $$ \Delta A_n = -2\Im(\bar{u}_n(\nabla -iA_n)u_n) =: -J(u_n,A_n)$$ and suppose that $u_n \in L^{\infty}(I,H^1(\Omega))$ for some bounded $I\subset \mathbb{R}$, $\Omega\...
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0answers
41 views

Fractional heat equation and analyticity

Consider the problem $$ \begin{cases} u_t + (-\Delta)^s u = 0 & \text{ in } \Omega \times (0,\infty) \\ u(x,t) = 0 & \text{ in } (\mathbb R^n \setminus \Omega ) \times (0,\infty) \\ u(\cdot,0) ...
1
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1answer
65 views

References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
2
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1answer
154 views

Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary

I'm interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $\Omega\subset\mathbb{R}^n$. It seems that the following statement can be deduced from known results:...
4
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1answer
167 views

Uniqueness of distributional solutions to the Poisson equation

Let $S(\mathbb R^n)$ denote the space of all Schwartz functions on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^*$ denote its dual. My ...
2
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0answers
41 views

Deducing elliptic regularity using spherical formula for Laplacian

It is well known that if $\Delta u = f$ and $u\in H_0^1(U),f \in L^2(U)$ for some domain $U,$ then we have $\|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}$ for some constant $C.$ One way to prove this is to ...
2
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1answer
58 views

How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?

I have encountered the following problem. Let $\chi:=\chi_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$. $\nabla\cdot ((1+\chi_{B(0,1/2)})\nabla u )=0 $ in $...
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0answers
160 views

Differential equation on a Riemannian manifold

Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\phi+h$, the decomposition of $\theta$ according to the Hodge decomposition. ...
1
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1answer
73 views

Poisson equation in a periodic strip

Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem $$-\Delta u=f$$ with boundary conditions $$...
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0answers
59 views

Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain

Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$. Consider the Laplace equation with a Neumann boundary condition $$ -\Delta u = 0 \quad\mbox{in } \Omega, \qquad \frac{\...
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1answer
67 views

Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
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0answers
48 views

Eigenvalues of the Laplacian and min-max formula in any space dimension

In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by $$ \lambda_1 = \min_{u \in H^1_0(\Omega), \|...
1
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0answers
108 views

Liouville theorem for elliptic equation with advection term

How can one prove that any $L^2$ solution of $$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$ is zero if $a(x)$ is a divergence-free vector field such that $\int |\...
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0answers
47 views

Fractional Sobolev embedding theorem

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
0
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1answer
57 views

Gehring Lemma in dimension 2

In Iwaniec's paper presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$. Question. What about dimension 2: can we actually go down to $p=1$?
1
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1answer
60 views

Fractional Laplacian equation on a ball and explicit solutions

Let us consider \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
1
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1answer
167 views

Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$

Let us consider the problem $$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$ where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and $v:\mathbb R^n \to \...
2
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0answers
49 views

Reference Request: Dirichlet operators with singular coefficients

Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by \begin{align*} \mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
0
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1answer
88 views

Existence of solutions of a system of first order PDEs

Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset. Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions. That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
3
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1answer
220 views

Find strictly subharmonic function that vanishes at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
4
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0answers
70 views

Gradient estimates

Gradient estimates (and especially the differential Harnack) for harmonic functions on Riemannian manifolds were proved by Cheng and Yau in 1975, by using Bochner's formula. However, it seems that ...
2
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0answers
46 views

Non-separable Laplace-Beltrami eigenfunctions have isolated critical points (reference request)

Consider the Laplace-Beltrami operator on a compact manifold. Generically, Uhlenbeck has shown that eigenfunctions of the Laplace-Beltrami operator are Morse functions. But there are some manifolds, ...
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0answers
88 views

Series and solution of $-\Delta u + \lambda u = f(x)$

Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of \begin{align*} -\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
3
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1answer
202 views

Solution of the fractional Laplace equation on a ball

What is the expression of the (non $u \equiv 0$) solutions to \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} ...
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1answer
72 views

Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]

Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
0
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2answers
96 views

Fractional laplacian of $(a-x)_+^\alpha$ in $(0,1)$

How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$? Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
3
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2answers
198 views

Gradient estimates for a boundary value problem

$\newcommand{\avint}{⨍}$ Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE: $$ \begin{cases} -\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\ w=0 \...
2
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2answers
99 views

Contours for harmonic functions in bounded domains

I was curious about researching different methods of solving Laplace's equation and was wondering if there is anything on generating curves on the domain where the solution is constant (contours). For ...
4
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1answer
62 views

Gauge fixing for a semi-relativistic model involving electromagnetism

When studying non-relativistic charged particles in an electromagnetic field with self-interaction on usually relies on the Schrödinger-Maxwell system \begin{align} i\partial_t u = -(\nabla-iA)^2 u + ...
3
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0answers
67 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 $\...
0
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1answer
204 views

Is this PDE solvable?

Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$: $$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text{...
0
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1answer
81 views

Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?

The following inequality is trivially true $$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
5
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1answer
240 views

A scaled fractional ''Sobolev inequality''

Does a fractional interpolation inequality similar to $$ \int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
1
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0answers
49 views

Integration by parts with fractional Laplacian and divergence

Let $\eta \in C^\infty_c$ and $\xi \in C^2$ Is it true that the integration by parts formula $$ \int_{\mathbb R^N} \nabla \eta \cdot \nabla((-\Delta)^{(\alpha-2)/2} \xi) = \int_{\mathbb{R}^N} \nabla \...
5
votes
1answer
119 views

Mean value principle reversed

Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique ...
-1
votes
1answer
60 views

Regularity of stationary incompressible Navier-Stokes equations in $\mathbb R^2$

What is the regularity of solutions for the stationary incompressible Navier-Stokes equations \begin{align*} -\Delta u +u\cdot \nabla u + \nabla p &= 0\\ \nabla \cdot u &= 0 \end{align*} in $\...
3
votes
1answer
164 views

Alternative proof of Liouville theorem for harmonic functions

From Prove Liouville theorem without using mean value property the following question arises: To prove the Liouville theorem If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
3
votes
1answer
118 views

Duality argument for elliptic regularity

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
4
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0answers
72 views

Fractional Laplacian and chain rule

For the classical Laplacian, we have $$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$ for smooth functions $h$ and $u$. Does a similar chain rule hold (up to a reminder term) also for the ...

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