# 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.

761
questions

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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 ...

**0**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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{...

**6**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...

**0**

votes

**0**answers

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
$G_{\lambda,s}=(\Delta^s+\lambda)^{-1}$. ...

**0**

votes

**0**answers

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**

votes

**1**answer

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:...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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\...

**0**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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 $...

**5**

votes

**0**answers

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**

vote

**1**answer

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
$$...

**0**

votes

**0**answers

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{\...

**5**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

vote

**0**answers

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 |\...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

vote

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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, ...

**1**

vote

**0**answers

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**

votes

**1**answer

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*}
...

**-1**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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 ...