**2**

votes

**2**answers

112 views

### What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...

**1**

vote

**0**answers

52 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**

vote

**0**answers

43 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

82 views

### Regularity of weak solution

I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...

**0**

votes

**1**answer

35 views

### References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity.
I studied always, following Evans book "PDE", the case with ...

**3**

votes

**3**answers

145 views

### Limits for eigenvalues for the Dirichlet Laplacian

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on ...

**8**

votes

**1**answer

222 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 ...

**1**

vote

**0**answers

70 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

39 views

### Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type
$$
-a\Delta u + f(u) = 0,
$$
$$
u|_\Gamma = u_0
$$
by Newton’s method when its convergence is global and monotonic.
...

**1**

vote

**0**answers

31 views

### Degenerate Carleman Estimate for Laplace Beltrami Operator

Suppose $(M,g)$ is a compact Riemannian manifold with boundary and that I have been able to prove a Carleman type estimate of the following form for an explicit phase function $\phi$ :
$ \| e^ {\tau ...

**0**

votes

**1**answer

91 views

### Square Integrable Harmonic Functions in an Infinite Strip

Suppose $S= \left\{x \in \mathbb{R}^3 : a <x_1< b \right\} $ is an infinite strip the three dimensional Euclidean Space.
Is it true that the only $L^2$ harmonic function in this strip is the ...

**0**

votes

**2**answers

102 views

### Operator on a Sobolev space [closed]

I'm studying Sobolev spaces using Evans' PDE book.
I can't figure out this simple fact.
Let $L$ be an operator in this form:
$$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$
I can't understand why ...

**1**

vote

**1**answer

145 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...

**3**

votes

**2**answers

396 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 ...

**2**

votes

**0**answers

49 views

### Uniformly continuity over Nemytskii functional in $D^{1,2}(\mathbb{R}^N)$

recently i am working in the following question:
Let $F(x,t)$ a Caratheodory function (i.e. $t \mapsto F(x,t)$ is continuous for all $x$ and $x \mapsto F(x,t)$ is Lebesgue mensurable for all $t$ ) ...

**3**

votes

**1**answer

630 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 ...

**0**

votes

**1**answer

97 views

### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
...

**2**

votes

**1**answer

63 views

### solvability of linear elliptic pde on a torus

Consider a linear non-divergent form elliptic PDE on a flat torus $\mathbf{T}^n$, $$a_{ij}\partial_{ij}u+b_i\partial_iu=f$$
where all the coefficients and $f$ are smooth. What is the condition that ...

**1**

vote

**0**answers

65 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**

vote

**0**answers

87 views

### Basic doubt in a free boundary problem for the Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf
In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...

**0**

votes

**0**answers

71 views

### Limit for eigenvalues of the Dirichlet problem

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, ...

**4**

votes

**1**answer

93 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 ...

**1**

vote

**1**answer

91 views

### Norm equivalent to Sobolev norm? [closed]

On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...

**5**

votes

**1**answer

116 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 $\partial\Omega$. ...

**4**

votes

**1**answer

118 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 ...

**1**

vote

**0**answers

80 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 ...

**3**

votes

**0**answers

92 views

### Regularity in PDE theory

I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...

**1**

vote

**0**answers

92 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 ...

**2**

votes

**1**answer

218 views

### Does this linear elliptic equation have a weak solution?

Let $Q = \Omega \times (0,C)$ where $\Omega$ is a bounded domain, write $(x,y) \in Q$ for $x \in \Omega$ and $y \in (0,C)$. Is the problem
$$\Delta_{(x,y)}v = 0\quad\text{in $Q$}$$
$$\frac{\partial ...

**1**

vote

**2**answers

140 views

### Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution ...

**3**

votes

**0**answers

125 views

### On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and ...

**1**

vote

**0**answers

52 views

### Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...

**1**

vote

**0**answers

30 views

### Eigenvalue overdetermined problem

Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$:
$$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\
u = 0 ...

**5**

votes

**0**answers

298 views

### Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the smooth tame Frechet space of smooth symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the smooth tame ...

**1**

vote

**0**answers

117 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 ...

**3**

votes

**1**answer

285 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 ...

**1**

vote

**0**answers

107 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 ...

**2**

votes

**0**answers

94 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 ...

**3**

votes

**1**answer

98 views

### Laplacian on non-compact domains

I consider non-compact domains $\Omega$ with cylindrical ends. For example, $\Omega$ has a cover $\Omega_0 \cup (0,\infty) \times M$, where $\Omega_0$ has finite measure and $M$ is a compact manifold.
...

**2**

votes

**2**answers

175 views

### “C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation
$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...

**0**

votes

**1**answer

98 views

### Nontrivial solutions of a semilinear elliptic equation

What is known, for $N\geq3$, about the existence of nontrivial real-valued solutions $u=u(x)$ of the following semilinear elliptic equation:
$$
\left\{ \enspace
\begin{aligned}
&\Delta u = f(u) ...

**0**

votes

**0**answers

50 views

### regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies:
\begin{equation*}
b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...

**2**

votes

**0**answers

50 views

### Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...

**1**

vote

**0**answers

79 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**

votes

**1**answer

147 views

### Analysis of Sobolev spaces [closed]

I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...

**2**

votes

**0**answers

126 views

### Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...

**0**

votes

**0**answers

55 views

### Control of Hessian by its trace in a bounded domain

We know if $u:\mathbb{R}^n \to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by parts, that
$$
\|\Delta u\|=\|\nabla^2u\| \tag 1
$$
where $||\cdot||$ is ...

**0**

votes

**0**answers

97 views

### Schauder estimate on a bounded domain

We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that
$$
||\Delta u||=||{\nabla}^2u||
$$
where $||\cdot||$ is the ...

**-1**

votes

**1**answer

101 views

### Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...

**5**

votes

**3**answers

363 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, ...