Tagged Questions
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1
vote
0answers
32 views
Gaussian curvature of conformal transformations
Let $g$ be a smooth metric and $g'=e^{v}g$, where $v$ is also a smooth function. Then it is well-known that
$(*) -\Delta_g u +2k_g=2 k_{g'}e^{v}$,
where $k_g$ and $k_{g'}$ are the Gaussian ...
1
vote
1answer
19 views
Generation of strictly contraction Semigroups
Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
1
vote
0answers
33 views
Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
2
votes
2answers
126 views
+50
Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
2
votes
0answers
93 views
DeGiorgi oscillation lemma
Where can I find a proof of the following result?
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(...
3
votes
1answer
108 views
Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
1
vote
0answers
58 views
Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary
In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669
Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...
2
votes
0answers
37 views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
4
votes
0answers
187 views
Spectral Gap of Elliptical Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of elliptical operator $ \nabla \cdot(a(x)\nabla)$ defined on $D$, can be controlled?
The boundary condition is that the solution at ...
5
votes
3answers
191 views
about the Hausdorff dimension of Removable singularities of PDE
There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let f ...
0
votes
0answers
67 views
Green functions for circular sectors
I would like to solve the Dirichlet boundary value problem
$$ (\Delta-k^2)u=0 \ \ \ \text{in $\Omega$} \\ u=f \ \ \ \text{on $\partial \Omega$} $$
where $\Omega$ is an infinite circular sector of ...
4
votes
2answers
214 views
Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)
There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs,
where $P_k$ is given by
\begin{equation}
\begin{aligned}
P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
5
votes
0answers
109 views
Extension of elliptic complex to an exact sequence
This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.
Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
0
votes
0answers
24 views
Eigenfunction of Distorted Laplacian on Smooth compact domain
Setup
Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...
1
vote
0answers
47 views
Solving Oblique Boundary value problem using Neumann Boundary Condition
\begin{align*}
&\frac{1}{2}a_{ij}(x)\frac{\partial^{2}u }{\partial x_{i}\partial x_{j}}+ b_{i}(x)\frac{\partial u}{\partial x_{i}}+ c(x)u=\lambda u ~~~in ~~\Omega\\
&\frac{\partial u}{\partial ...
0
votes
0answers
65 views
Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain
I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...
6
votes
1answer
197 views
is signed distance function real analytic for real analytic domains
If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by
\begin{equation}
f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\...
5
votes
1answer
102 views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
0
votes
0answers
50 views
Convergence in integral
I tried to prove that:
(f1) $f(0)=0$, $f\in C^1(0,\infty)$, $f'(s)>0$ for $s>0$, and there is a $p\in(2, 2N/(N-4))$ such that $\lim_{s\to \infty}f(s)/s^{p-2}<\infty$,
(f2) $F(s)=\int_0^sf(t)...
2
votes
1answer
111 views
Evans-Krylov theorem
Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
0
votes
0answers
46 views
Degeneracy of critical points of solutions to second order elliptic PDEs
Problem: Suppose that $Lu = f(x)$ on a strictly convex domain $\Omega\subset\mathbb{R}^2$ and $u=0$ on $\partial\Omega$ where $L$ is elliptic and second order. I am trying to determine when the ...
0
votes
0answers
17 views
Are solutions to the second order elliptic PDE $Lu=f(u,\nabla u, x)$ locally $L$-harmonic homogeneous polynomials?
These two papers by Caffarelli and Friedman [1], [2] show that a solution $u$ to the elliptic PDE
$$
\Delta u = f(u,\,\nabla u,\, x)\,,\quad \text{ on } \Omega\subset\mathbb{R}^n
$$
is locally a ...
3
votes
0answers
66 views
Anderson Localization and Homogenization theory
I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.
The question is mostly related to homogenization theory in mathematical physics.
$\textbf{...
1
vote
1answer
103 views
Existence of unique critical points to second order elliptic PDEs
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
0
votes
0answers
37 views
Elliptic Dirichlet BVP's for regions with multiple boundary components
Apologies for the vague title, it was getting rather long so I decided to just explain more in the body of the text.
I am curious about the state of understanding for existence and uniqueness of ...
1
vote
1answer
114 views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...
1
vote
0answers
34 views
First eigenvalue for domains in hyperbolic space
I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
0
votes
0answers
103 views
Analytical solve for this 4th order Linear PDE
I want to Solve analyticaly or numericaly a biharmonic equation with homogeneous boundary conditions homogeneous in rectangular domain [0,a]*[0,b]?
I considered its solution using Fourier analysis (...
3
votes
1answer
147 views
Does the space of harmonic forms change continuously with the metric?
Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...
0
votes
0answers
28 views
Change of variable for the Stokes equations
I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741).
Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...
2
votes
1answer
156 views
On the Calabi-Yau conjecture for minimal surfaces
Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded.
Is it possible to remove the assumption "finite ...
1
vote
0answers
44 views
Unclear inequality of L2 norms (Poisson equation for modeling flow)
I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
3
votes
2answers
234 views
Can we specify the value of harmonic forms at a point?
Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.
Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.
Does there exist an open ...
6
votes
1answer
211 views
Reference Question: Boundary Value Problem for Dirac Operator on Manifold with a non-smooth boundary
I am trying to find references for the following boundary value problem: Assume that $\Omega$ is a compact 3-dim spin manifold with Dirac operator $D$ such that the boundary consists of two smooth ...
7
votes
0answers
78 views
Counter-examples to the higher dimensional statement of the half-space theorem
The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$.
The higher dimensional ...
2
votes
1answer
153 views
Reference request for weak solutions of an Elliptic PDE
Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one.
I want to find weak, non trivial, continuous, solutions of $$\...
3
votes
1answer
70 views
PDE satisfied by projection of a function onto a subspace
Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...
2
votes
0answers
44 views
Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$
I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$.
In the ...
1
vote
2answers
70 views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
0
votes
1answer
67 views
Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
3
votes
0answers
83 views
metric with curvature bounded in $L^2$
My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates ...
1
vote
1answer
89 views
Comparison principle for viscosity solution
I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen.
https://projecteuclid.org/euclid.jdg/1090349447
I ...
0
votes
0answers
37 views
Value at $(0,0)$ of the solution to a 2D linear elliptic PDE
The setup is the following.
Let $D=(-\frac{1}{2} , \frac{1}{2})^2 \subset \mathbb{R}^2$, $\partial D$ be its boundary. Let $a, b\in \mathbb{R}$ and $f$ be continuous on $\partial D$. Also we assume ...
0
votes
0answers
29 views
Unstable convergence of a Poisson equation
What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
1
vote
0answers
48 views
Conformal factors and light rays
Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $...
1
vote
1answer
166 views
Classification of a system of two second order PDEs with two dependent and two independent variables
If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
0
votes
2answers
126 views
Solving the Poisson equation using a random walk on $\mathbb Z ^d$
How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
0
votes
0answers
54 views
Continuity of solution to 2nd Order PDE w.r.t. the coefficients
I am considering the following 2nd order PDE :
On a domain $R$, for some $r > 0$,
\begin{equation*}
\frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...
1
vote
0answers
43 views
Regularity for Elliptic equation with mixed boundary condition in one dimensional domain [0,1]
I am stuck at establishing regularity for elliptic equations on the one dimensional domain $\Omega=[0,1]$.
The problem is
$Lu=f$, in $\Omega$, and $u(0)=0, u_x(1)=0.$
In the page 317, Evans' book, ...
10
votes
3answers
491 views
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
