**2**

votes

**0**answers

231 views

### Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...

**2**

votes

**0**answers

226 views

### Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary ...

**1**

vote

**0**answers

46 views

### Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...

**1**

vote

**0**answers

28 views

### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...

**1**

vote

**0**answers

69 views

### Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...

**1**

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**0**answers

56 views

### A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...

**1**

vote

**0**answers

135 views

### On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $

This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) ...

**1**

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**0**answers

38 views

### Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem
...

**1**

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**0**answers

47 views

### method for global existance for the NLS

We consider the nonlinar Schr\"odinger equation(NLS):
$$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$
where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...

**1**

vote

**0**answers

57 views

### Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...

**1**

vote

**0**answers

67 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

**1**

vote

**0**answers

24 views

### Regularity of a flux induced by a potential

Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a ...

**1**

vote

**0**answers

71 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

**1**

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**0**answers

42 views

### PDE Parameter-Dependent Center Manifolds

In the book 'Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems' by Mariana Haragus parameter-dependent center manifolds are discussed. Here it is assumed ...

**1**

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**0**answers

89 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

**1**

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**0**answers

72 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & ...

**1**

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**0**answers

80 views

### Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let ...

**1**

vote

**0**answers

54 views

### Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...

**1**

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**0**answers

55 views

### Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...

**1**

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**0**answers

74 views

### Examples for differential operators first order

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function ...

**1**

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**0**answers

48 views

### Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...

**1**

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**0**answers

71 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 , ...

**1**

vote

**0**answers

50 views

### The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...

**1**

vote

**0**answers

81 views

### Existence of the solution of a Dirichlet type differential equation

I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question:
...

**1**

vote

**0**answers

70 views

### Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by:
$$
...

**1**

vote

**0**answers

91 views

### Regularity result for the elliptic equation with Neumann conditions

I am having troubles to justify well some inequalities related with the classical
theory of elliptic equations. Let us consider the problem
\begin{align*}
-\Delta c & =f,\text{ in }\Omega\\
...

**1**

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**0**answers

61 views

### Heat equation inequality

There is an inequality that tells us that for some sufficiently smooth $f$ satisfying $(\partial_t - \Delta )f \le - \delta f^2 +K$ for $\delta,K >0$ that $f$ is bounded by some constant. ...

**1**

vote

**0**answers

128 views

### Gilbarg-Trudinger's book Theorem 4.13

I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...

**1**

vote

**0**answers

94 views

### Asymptotics of “heat” semigroup

Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...

**1**

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**0**answers

80 views

### Energy inequalities for Sobolev spaces of negative integer

I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| ...

**1**

vote

**0**answers

72 views

### Sobolev trace of $H^1(\mathcal{M} \times I)$ functions

Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times ...

**1**

vote

**0**answers

110 views

### Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
...

**1**

vote

**0**answers

55 views

### Equivalence of two definitions of weak solution (subtlety with null sets)

Consider
$$y_t - \Delta y = f$$
$$y(0) = y_0$$
with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions:
We have $y \in ...

**1**

vote

**0**answers

112 views

### $L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...

**1**

vote

**0**answers

51 views

### sharp conditions characterizing the vanishing of scalar Jacobi fields

Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$.
I was ...

**1**

vote

**0**answers

39 views

### Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...

**1**

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**0**answers

59 views

### persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...

**1**

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**0**answers

107 views

### maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...

**1**

vote

**0**answers

70 views

### Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to
$$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$
for some elliptic differential operator $L$ where $f$ is not ...

**1**

vote

**0**answers

99 views

### $H=W$ for weighted Sobolev spaces

Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights?
Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm
...

**1**

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**0**answers

64 views

### Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...

**1**

vote

**0**answers

126 views

### Strichartz estimates for the wave equation

Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...

**1**

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**0**answers

56 views

### Isolated singularities of harmonic mappings

Can a homeomorphic harmonic mapping $f=(u,v,w):\Omega\to \Omega'$ have isolated singular points. Here $\Delta f =0$, and singular point is a point with zero Jacobian. This will extend Lewy theorem for ...

**1**

vote

**0**answers

73 views

### Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...

**1**

vote

**0**answers

122 views

### Harmonic extension of $L^\infty$ function is in $L^\infty$?

Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = ...

**1**

vote

**0**answers

41 views

### Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?

**1**

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**0**answers

196 views

### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**1**

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**0**answers

59 views

### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

**1**

vote

**0**answers

139 views

### Help in understanding “Local well-posedness for the Maxwell-Schrodinger system”

Is there someone who knows the following paper
"Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada.
I'm trying to study it but I've some doubts. In particular I'm not ...

**1**

vote

**0**answers

107 views

### Showing existence of positive weak solution of a PDE by CoV

Given the following PDE
$$
\begin{cases}
-\Delta u+\alpha=u^q &x\in\Omega\\
u=0 &x\in\partial\Omega
\end{cases}
$$
where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...