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
1
vote
0answers
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
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 $\mathbb R^d$
$$
\begin{...
3
votes
1answer
83 views
Reaction-diffusion systems treated as dynamical systems
I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems.
I have the book of Alain Haraux – Systèmes dynamiques dissipatifs et ...
2
votes
1answer
77 views
Reference request on Pucci extremal operators
While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by:
$$M_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{...
6
votes
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
votes
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
votes
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 $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
0
votes
0answers
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
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
votes
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 $H^2(B_1)\cap H^1_0(B_1)\to L^2(B_1)$
$$T:...
1
vote
0answers
40 views
Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?
Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
1
vote
0answers
28 views
Strong convergence from elliptic equation
Consider the following equation
$$ \Delta A_n = -2\Im(\bar{u}_n(\nabla -iA_n)u_n) =: -J(u_n,A_n)$$
and suppose that $u_n \in L^{\infty}(I,H^1(\Omega))$ for some bounded $I\subset \mathbb{R}$, $\Omega\...
0
votes
0answers
41 views
Fractional heat equation and analyticity
Consider the problem
$$
\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
vote
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
votes
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
votes
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
votes
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
votes
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 $...
5
votes
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
vote
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
$$...
0
votes
0answers
59 views
Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$.
Consider the Laplace equation with a Neumann boundary condition
$$
-\Delta u = 0 \quad\mbox{in } \Omega, \qquad
\frac{\...
5
votes
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 ...
0
votes
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
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
1
vote
0answers
108 views
Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
1
vote
0answers
47 views
Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
0
votes
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
vote
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 $$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
1
vote
1answer
167 views
Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$
Let us consider the problem
$$ (-\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
votes
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
votes
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
votes
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
votes
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
votes
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, ...
1
vote
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
votes
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*}
...
-1
votes
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.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
0
votes
2answers
96 views
Fractional laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
3
votes
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:
$$
\begin{cases}
-\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\
w=0 \...
2
votes
2answers
99 views
Contours for harmonic functions in bounded domains
I was curious about researching different methods of solving Laplace's equation and was wondering if there is anything on generating curves on the domain where the solution is constant (contours). For ...
4
votes
1answer
62 views
Gauge fixing for a semi-relativistic model involving electromagnetism
When studying non-relativistic charged particles in an electromagnetic field with self-interaction on usually relies on the Schrödinger-Maxwell system
\begin{align}
i\partial_t u = -(\nabla-iA)^2 u + ...
3
votes
0answers
67 views
Reference requests: $W_{p}^1$-estimate for $(\triangle -\lambda)$ on Lipschitz domains
Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain
with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\...
0
votes
1answer
204 views
Is this PDE solvable?
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is unit ball. I am trying to solve the following PDE for $f$:
$$\Delta f -\frac{ f }{r^2}+ \frac{ \left. f \right|_{\partial M}}{r^2} = 0, \qquad \text{...
0
votes
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
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
5
votes
1answer
240 views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\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
vote
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
$$
\int_{\mathbb R^N} \nabla \eta \cdot \nabla((-\Delta)^{(\alpha-2)/2} \xi) = \int_{\mathbb{R}^N} \nabla \...
5
votes
1answer
119 views
Mean value principle reversed
Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique ...
-1
votes
1answer
60 views
Regularity of stationary incompressible Navier-Stokes equations in $\mathbb R^2$
What is the regularity of solutions for the stationary incompressible Navier-Stokes equations
\begin{align*}
-\Delta u +u\cdot \nabla u + \nabla p &= 0\\
\nabla \cdot u &= 0
\end{align*}
in $\...
3
votes
1answer
164 views
Alternative proof of Liouville theorem for harmonic functions
From Prove Liouville theorem without using mean value property the following question arises:
To prove the Liouville theorem
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
3
votes
1answer
118 views
Duality argument for elliptic regularity
M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
4
votes
0answers
72 views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\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 ...
