The regularity tag has no usage guidance.

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### elliptic regularity of Neumann problem on Square

I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) ...

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56 views

### elliptic regularity for Neumann BVP on square

I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ ...

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61 views

### biharmonic equation with L^1 data and Navier Condition

I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...

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**1**answer

240 views

### Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded ...

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137 views

### Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...

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**1**answer

291 views

### Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...

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99 views

### About a classic result from Han and Lin's PDE book

Let $A=a_{ij}$ be an $n \times n $ a symmetric matrix where the coeficients are in $L^{\infty}(B_r(0))$ and satisfies
$$ \lambda |\xi|^2 \leq a_{ij}(x)\xi_i\xi_j \leq \alpha |\xi|^2, \ x \in ...

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**1**answer

125 views

### Distance function from a topological submanifold

Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can ...

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**1**answer

76 views

### Regularity of solution to a hyperbolic pde

I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^2(0,T;V^*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a ...

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**1**answer

167 views

### On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...

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**1**answer

88 views

### Approximation of sets by sets with regular border

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...

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**1**answer

89 views

### Topological description of the regular values of a differentiable function

Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms?
In particular, is the set of regular values ...

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26 views

### Capacity approximations by sets with regular boundary

Suppose I have a continuous, compactly supported function $f : \mathbb{R}^2 \to \mathbb{R}_{+}$ and I define the set $S := f^{-1}([a,\infty)) \subset \mathbb{R}^2$ for some $a > 0$. It is a ...

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**1**answer

245 views

### Does the implicit function theorem hold for discontinuously differentiable functions?

(This was posted on math.SE over 5 days ago and has not been answered,
although a comment mentioned a similar question on this site.)
Wikipedia's statement of the implicit function theorem requires ...

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**1**answer

108 views

### W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞}

Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$.
Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$?
...

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164 views

### Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...

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**5**answers

1k views

### Are all almost regular graphs obvious?

Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.
A graph is almost regular if $\Delta-\delta=1$.
Now, here is a simple way to generate ...

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**0**answers

87 views

### mixed Dirichlet Neumann regularity for an elliptic equation

Here is a problem which may be easy for some of you but not for me.
Statement of the problem:
Denote $\Omega := \{ (x,y) \in (0, \infty) \times (-\infty,\infty) \}$.
Let $f \in L^2(\Omega)$ then by a ...

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votes

**1**answer

80 views

### Optimal Regularity for Invariance of Curvature under Isometries

It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...

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**2**answers

304 views

### Physical and real life interpretation of the concept of regularity used in differential equations?

I guess the title kind of speaks for my questions: I'm curious to know what could be the physical interpretation or real life application of the concept of regularity that arises in PDE: take for ...

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**1**answer

235 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 ...

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**0**answers

83 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 \, ...

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**3**answers

492 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 ...

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**0**answers

181 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 < ...

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**0**answers

87 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 ...

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**1**answer

553 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. ...

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**1**answer

92 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 ...

**3**

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134 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 ...

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**1**answer

210 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 ...

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votes

**1**answer

142 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 ...

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152 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 ...

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**1**answer

200 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 ...

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**2**answers

124 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 ...

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**4**answers

716 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 ...

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votes

**2**answers

321 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

**1**answer

522 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 ...

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votes

**1**answer

286 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 ...

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**2**answers

645 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 ...

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**3**answers

432 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) ...

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**1**answer

743 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

**1**answer

425 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 ...

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**0**answers

142 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 ...

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votes

**2**answers

554 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 ...

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votes

**1**answer

285 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 $\Bbb{R}^n$, ...

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**0**answers

288 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 ...

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**2**answers

516 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 ...

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**0**answers

160 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 ...

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**1**answer

490 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 ...

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222 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 ...

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**1**answer

332 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 ...