**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 ...

**16**

votes

**1**answer

524 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 ...

**14**

votes

**4**answers

943 views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**13**

votes

**4**answers

888 views

### Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...

**12**

votes

**2**answers

483 views

### Does the Legendre-Hadamard condition imply a generalized Gårding inequality?

For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
...

**10**

votes

**1**answer

199 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

**9**

votes

**1**answer

240 views

### What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?

We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$?
The $p$-Laplacian ...

**8**

votes

**3**answers

444 views

### How to define the square root of $1-\Delta $?

If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...

**8**

votes

**1**answer

202 views

### Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...

**7**

votes

**1**answer

2k views

### Yau's conjecture for positive Chern class

I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this ...

**7**

votes

**2**answers

362 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 ...

**7**

votes

**2**answers

177 views

### Is there a maximum principle for stress in continuum mechanics?

I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...

**7**

votes

**1**answer

175 views

### Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...

**7**

votes

**0**answers

796 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 ...

**6**

votes

**2**answers

2k views

### Moser iteration for elliptic systems

I heard that De Giorgi-Nash-Moser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there ...

**6**

votes

**3**answers

1k views

### Betti number and harmonic forms

Dear Experts,
On a compact, boundless, Riemannian manifold, the dimension of the space of harmonic k-forms is equal to the k-th Betti number. Is this correct (by Hodge theory)? For example, on the ...

**6**

votes

**3**answers

631 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 ...

**6**

votes

**3**answers

192 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 ...

**6**

votes

**1**answer

183 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**6**

votes

**2**answers

1k views

### Estimates on the Green function of an elliptic second order differential operator.

Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...

**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 ...

**5**

votes

**1**answer

526 views

### Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one.
$I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...

**5**

votes

**3**answers

442 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) ...

**5**

votes

**1**answer

702 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 ...

**5**

votes

**3**answers

993 views

### Integration by parts for a general negative-definite self-adjoint operator.

I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...

**5**

votes

**1**answer

344 views

### Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?

Context: I am currently reading through the freely available lecture notes from Tristan Riviere (here) on the applicability of integration by compensation in the analysis of various geometrically ...

**5**

votes

**3**answers

364 views

### Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...

**5**

votes

**1**answer

207 views

### Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$

A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...

**5**

votes

**1**answer

125 views

### significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...

**5**

votes

**1**answer

255 views

### Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...

**5**

votes

**1**answer

60 views

### Modified mean value property

Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant.
I met this modified mean value property of a solution $u$ of $Lu=0$ as
...

**5**

votes

**1**answer

164 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

**5**

votes

**1**answer

368 views

### A moving boundary in rock mechanics

I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...

**5**

votes

**0**answers

293 views

### Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...

**4**

votes

**2**answers

225 views

### A question on certain elliptic PDE

Consider the elliptic PDE "CR"
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are similar to the Cauchi ...

**4**

votes

**2**answers

917 views

### The Hölder continuity condition of the Schauder estimates

The classical Schauder estimates (see the link)
http://en.wikipedia.org/wiki/Schauder_estimates
Requires $f\in C^\alpha$ in order to get a solution $u\in C^{2+\alpha}$ of the equation
$$\Delta u=f$$
...

**4**

votes

**2**answers

272 views

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

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

**4**

votes

**1**answer

300 views

### Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...

**4**

votes

**2**answers

368 views

### Can the solution of an elliptic operator with smooth coefficients have zeros of infinite order?

I've been told that the answer is no, but I'm having a hard time finding a reference. More precisely, I'm interested in the following. Let $\Omega \subset \mathbb{R}^n$ be a bonded open connected set, ...

**4**

votes

**2**answers

569 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**4**

votes

**1**answer

328 views

### A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two ...

**4**

votes

**1**answer

101 views

### An inequality for eigenvalues of the Dirichlet problem

Is either of these inequalities true?
$$\lambda(tA + (1-t)B)\geq t\lambda(A) + (1-t)\lambda(B)$$
or
$$\lambda(tA + (1-t)B)\leq t\lambda(A) + (1-t)\lambda(B),$$
where $0\leq t \leq 1$, $A,B$ are ...

**4**

votes

**1**answer

212 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 ...

**4**

votes

**1**answer

72 views

### Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...

**4**

votes

**1**answer

179 views

### Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...

**4**

votes

**1**answer

242 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 ...

**4**

votes

**1**answer

153 views

### Gradient estimate for elliptic equation

Given:
1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$
2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$
3)$g=(g_i,\ldots,g_n)\in ...

**4**

votes

**1**answer

246 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**4**

votes

**1**answer

621 views

### regularity of solutions of fractional laplacian

Hello, I am looking for boundary regularity of solutions of $(-\Delta)^s u= f(x)$ in $\Omega$ with Dirichlet boundary conditions and where $f $ is nice enough say $f\in C^{1,\alpha}(\overline\Omega)$. ...

**4**

votes

**0**answers

58 views

### Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
...