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0
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1answer
63 views

reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
0
votes
0answers
35 views

Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
1
vote
2answers
320 views

Elliptic theory on compact manifolds

Maybe this is silly. On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation $$ \Delta u=f \quad\text{ in $\Omega$}$$ $$ u=0\quad\text{ on $\partial\Omega$}.$$ One has the following ...
1
vote
0answers
106 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
2
votes
0answers
77 views

Constant in Maximal sobolev regularity

We know the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach ...
4
votes
1answer
207 views

What's wrong with the Courant nodal domain theorem

The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. ...
-1
votes
1answer
77 views

Regularity of solutions for a non linear elliptic equation

Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$ $(-\Delta)^2 v_k=e^{v_k}$ $v_k(x)\leq v_k(0)=0$ $\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad ...
2
votes
0answers
70 views

Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain

Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
2
votes
1answer
155 views

Proof of regularity for bounded elliptic problem

We consider the boundary value problem for potential in the form: $$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$ with boundary conditions $$\nabla ...
2
votes
1answer
98 views

sub and super-levelset regularity for Sobolev functions

I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely: Assume ...
2
votes
0answers
135 views

Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules

Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on ...
3
votes
0answers
96 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 ...
2
votes
2answers
112 views

Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
10
votes
4answers
428 views

Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
3
votes
2answers
232 views

Non symmetric coefficient matrix for elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form $$ D_i(a_{i,j}D_ju)=0 \tag{1}$$ where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
5
votes
1answer
461 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 ...
3
votes
1answer
167 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 ...
0
votes
2answers
505 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 ...
5
votes
3answers
337 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
1answer
393 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
6
votes
1answer
302 views

If a compact Kahler manifold $(M,g)$ has constant scalar curvature, is the metric $g$ real analytic?

Hi to all! Perhaps it is a silly question, if so i'll delete this post. Suppose we have a compact Kahler manifold $(M,g)$ of complex dimension $m$ with constant scalar curvature with respect to its ...
2
votes
0answers
123 views

regularity for viscosity solutions of second order parabolic equations

I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre . Here F is ...
2
votes
2answers
376 views

Does regularity of the boundary imply interior sphere condition

In the article of Massari presented here there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition: There exists $\rho>0$ such that for every ...
0
votes
1answer
253 views

A property of sets of finite perimeter

I have a question regarding sets of finite perimeter. I feel that it should be true, but I didn't manage to prove or find a reference about it. Suppose $D$ is an open, bounded subset of $\Bbb{R}^n$, ...
6
votes
0answers
265 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...
4
votes
2answers
418 views

Boundary regularity for the Dirichlet problem

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator. We wish to solve the Dirichlet problem ...
2
votes
0answers
156 views

Manifolds with a lower degree of regularity

Hello guys, I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below). There, the authors consider domains of $R^n$ with regularity of class $W^2 ...
7
votes
1answer
392 views

What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
3
votes
0answers
215 views

Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...
2
votes
1answer
291 views

What is the regularity of the argument of a complex function?

Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
1
vote
2answers
652 views

Regular vs. Irregular Vertices in a Mesh

Hi everybody, Reading about Geometry Processing, I have realized that people in this area are very interested in regular vertices(degree=6) rather than irregular ones. Can anybody give me reasons ...
4
votes
2answers
1k 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 ...