**2**

votes

**0**answers

98 views

### Regularity of solution of nonlinear equation

Hi!
Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball
$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.
Let $f\in C^{0,\alpha}(B_1)$ ...

**7**

votes

**2**answers

331 views

### Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...

**4**

votes

**1**answer

218 views

### Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...

**1**

vote

**1**answer

259 views

### Regularity of the right hand side (the source term) in Evans-Krylov theory

A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...

**2**

votes

**1**answer

135 views

### The maximum in the Poisson problem on the cube with constant source

Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial ...

**0**

votes

**1**answer

204 views

### Mean value theorem for harmonic functions on ellipsoid

Is there any result like the mean value theorem for harmonic functions on ellipsoids (instead of sphere)?

**0**

votes

**1**answer

140 views

### H_0^1 and C_infinity on the interior, does that imply classical limit is 0 on the boundary?

The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H_0^1 and also belong to C_infinity on the interior. Does that mean the classical limit of the function ...

**1**

vote

**0**answers

148 views

### Result like Brezis-Kato Lemma for Biharmonic equation.

Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\ $\Delta^2 u = a(x)u$ and $a(x)\in L^{\frac{n}{4}}(\Omega)$, where $\Omega$ is bounded. If ...

**15**

votes

**1**answer

490 views

### Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...

**1**

vote

**1**answer

222 views

### In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation
$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...

**0**

votes

**1**answer

136 views

### Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: ...

**0**

votes

**2**answers

626 views

### Interior regularity for elliptic equations

The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...

**3**

votes

**1**answer

317 views

### why is it called the diamagnetic inequality?

How is the Diamagnetic inequality born? Why is it call this name?
Diamagnetic inequality： $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.

**5**

votes

**1**answer

1k views

### Nash's paper on parabolic equations.

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND
ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958). The author there establishes some a ...

**1**

vote

**1**answer

98 views

### Boundedness of a given boundary value problem.

I've been given the following BVP:
\begin{align*}
-\Delta u = u- u^3,\: x\in \Omega
\end{align*}\begin{align}
u = 0,\: x\in \partial \Omega
\end{align}
where $\Omega\subset \mathbb{R}^N$ is bounded.
...

**0**

votes

**1**answer

218 views

### Quantitative Global Schauder Estimates/Hölder Regularity

Consider the linear second order elliptic Dirichlet problem
$$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$
Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda ...

**3**

votes

**2**answers

399 views

### Schauder estimates for higher order linear elliptic operator on manifold

Hi!
Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and ...

**6**

votes

**0**answers

726 views

### elliptic regularity on manifolds

Hello!
I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic ...

**0**

votes

**0**answers

126 views

### second derivatives of initial conditions for a non-linear PDE

Hello, I have the following problem:
Having a non-linear PDE of second order, there are the initial conditions
$F(x,0)=g(x)$,
$\partial_yF(x,0)=h(x)$.
The question is: Can the second derivatives ...

**5**

votes

**3**answers

417 views

### Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...

**4**

votes

**2**answers

256 views

### T. Carleman's method on eigenvalues asymptotics

What is the best available less or more modern introduction to the subject?

**2**

votes

**1**answer

293 views

### Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int_{\Omega} \nabla u \nabla v d x = ...

**4**

votes

**1**answer

203 views

### Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...

**1**

vote

**2**answers

335 views

### Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...

**1**

vote

**1**answer

367 views

### Elliptic Differential Equations with rough boundary data

Question stated roughly:
Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...

**2**

votes

**1**answer

210 views

### Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation
\begin{equation}
-\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0.
...

**3**

votes

**1**answer

303 views

### Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...

**0**

votes

**1**answer

144 views

### What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...

**1**

vote

**1**answer

443 views

### Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...

**3**

votes

**2**answers

244 views

### Boundary Value Problem in the space of Distributions

I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the ...

**3**

votes

**2**answers

616 views

### Elliptic regularity on bad domain

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem
$$
-\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega.
$$
If $\varphi$ is a continuous function, ...

**1**

vote

**0**answers

140 views

### Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation
$$
-\triangle u + u + u^3 = g, \quad x \in R^3.
$$
If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...

**1**

vote

**1**answer

198 views

### Reference to the Existence and Uniqueness of the PDE system

Hi all
I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...

**2**

votes

**1**answer

240 views

### Uniform upper estimate for derivatives of solution of an elliptic PDE

Let for every $i=1,\dots,n$ and $j=1,\dots,n$
be given functions $a^{ij}\colon\mathbb{R}^n\to \mathbb{R}$, $b^{i}\colon\mathbb{R}^n\to \mathbb{R}$, $c\colon\mathbb{R}^n\to \mathbb{R}$ from space ...

**2**

votes

**1**answer

387 views

### Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...

**1**

vote

**1**answer

231 views

### Harmonic/conformal map composition between manifolds in either order?

Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is ...

**2**

votes

**2**answers

418 views

### How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...

**1**

vote

**1**answer

809 views

### Extension of harmonic function

Suppose $u$ is a harmonic function of a domain $\Omega\subset \mathbb{R}^n$ and $u$ is continuous up to the boundary. If $\partial\Omega$ has an open smooth portion, can $u$ be extended to a harmonic ...

**0**

votes

**1**answer

213 views

### Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving ...

**3**

votes

**1**answer

353 views

### A property of weakly harmonic functions

Let $u$ be a distribution in $\mathbb{R}^{n}$ and $\Delta u$=0 in the distributional sense. In addition $u\in L^{p}(\mathbb{R}^{n})$, $p>1$, then can we conclude that $u$ is zero almost everywhere ...

**6**

votes

**3**answers

181 views

### Stability of the spectrum for perturbations of the boundary

Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be ...

**2**

votes

**2**answers

304 views

### Reference for all solutions of homogeneous elliptic and parabolic equations with Hölder continuous coefficients to be classical

Consider a uniformly elliptic equation
$$
\sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u+\sum_{i=1}^n b_{i}(x)\partial_{i}u+c(x)u=0
$$
say, in an open ball $B\subset \mathbb R^n$, where coefficients are ...

**3**

votes

**1**answer

615 views

### Variational formulation for bilaplacian

I am trying to derive a variational formulation for the following problem $$\left\{ \begin{array}{ll} \Delta^2u=f, & \Omega \\ \Delta u+\rho \partial_{\nu}u=0, & \partial \Omega ...

**3**

votes

**2**answers

511 views

### Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $

The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ?
For detail :
Fix $p \geq 1. ...

**1**

vote

**2**answers

876 views

### Geometric Mean Value Property

Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...

**6**

votes

**3**answers

605 views

### Moser regularity proof avoiding John-Nirenberg lemma

I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely ...

**1**

vote

**0**answers

154 views

### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta ...

**2**

votes

**0**answers

189 views

### Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that ...

**2**

votes

**0**answers

162 views

### A first order ODE with Sobolev estimates

Suppose $f: {\mathbb R}\to {\mathbb R}$ is a continuous function satisfying
$$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T.$$
(For simplicity we may even assume that $f$ is ...

**19**

votes

**3**answers

2k views

### Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...