**6**

votes

**0**answers

162 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 ...

**6**

votes

**0**answers

136 views

### Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems:
Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...

**6**

votes

**0**answers

699 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 ...

**5**

votes

**0**answers

281 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 ...

**5**

votes

**0**answers

279 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

**0**answers

56 views

### How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in ...

**4**

votes

**0**answers

64 views

### A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting:
we have a non-negative supersolution $h$ of the following elliptic equation:
\begin{equation}
\Delta h + \|\nabla h\|^2 + f(x) \geq 0
...

**4**

votes

**0**answers

162 views

### introduction books for Dynamic systems of discrete Schrodinger operator for beginner

In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...

**4**

votes

**0**answers

79 views

### Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...

**4**

votes

**0**answers

460 views

### Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...

**4**

votes

**0**answers

327 views

### Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...

**3**

votes

**0**answers

90 views

### Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk $\Omega$ with subject to Robin boundary conditions. The formulation is as ...

**3**

votes

**0**answers

78 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 \, ...

**3**

votes

**0**answers

107 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 ...

**3**

votes

**0**answers

83 views

### A question about fractional integrals

I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...

**3**

votes

**0**answers

349 views

### problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution?
$$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...

**2**

votes

**0**answers

73 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 ...

**2**

votes

**0**answers

44 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 ...

**2**

votes

**0**answers

99 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 ...

**2**

votes

**0**answers

32 views

### Schauder-type estimates for polyharmonic operators in a smooth domain of $R^N$

Let $L$ be an elliptic operator of the form
$$
Lu := (-1)^m \sum_{|\alpha|=2m} a_\alpha(x) D^\alpha u + \sum_{|\alpha|\leq 2m-1} b_\alpha (x) D^\alpha u
$$
with smooth coefficients and $u$ defined ...

**2**

votes

**0**answers

56 views

### probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions:
Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...

**2**

votes

**0**answers

63 views

### Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.
I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...

**2**

votes

**0**answers

72 views

### The level set of convolution

Suppose $u\in C^2(\bar \Omega)\cap C^\infty(\Omega)$ is a function which is compact supported on $\Omega$ with $u\equiv 0$ at $\partial\Omega$, and $\Omega\subset \mathbb R^3$.
We also assume that ...

**2**

votes

**0**answers

216 views

### A integral equation with Discrete to result by inverse problem

Problem
I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...

**2**

votes

**0**answers

55 views

### Moser's iteration for non homogeneous quasilinear elliptic PDE

I want to know some reference of Moser's iteration for non homogeneous quasilinear uniformly elliptic PDE, in particular, $u_{ij}(\delta_{ij}+u_iu_j|\nabla u|^2)=0$. I want to know how to deal with ...

**2**

votes

**0**answers

80 views

### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...

**2**

votes

**0**answers

96 views

### Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$

Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...

**2**

votes

**0**answers

153 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 ...

**2**

votes

**0**answers

42 views

### reference on existence result for nonlinear elliptic PDE

During my work, I came to the question of existence of weak solutions to the following elliptic equation
$$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$
with ...

**2**

votes

**0**answers

96 views

### slightly subcritical elliptic pde; the linearized equations

Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems
$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...

**2**

votes

**0**answers

89 views

### A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...

**2**

votes

**0**answers

98 views

### Regularity of solution of nonlinear equation

Hi!
Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball
$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.
Let $f\in C^{0,\alpha}(B_1)$ ...

**2**

votes

**0**answers

186 views

### Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that ...

**2**

votes

**0**answers

162 views

### A first order ODE with Sobolev estimates

Suppose $f: {\mathbb R}\to {\mathbb R}$ is a continuous function satisfying
$$f(s) = c_- >0,\forall s<-T; f(s) = c_+<0, \forall s>T.$$
(For simplicity we may even assume that $f$ is ...

**2**

votes

**0**answers

287 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**2**

votes

**0**answers

251 views

### Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?

Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to ...

**2**

votes

**0**answers

296 views

### Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...

**2**

votes

**0**answers

284 views

### Poisson problem with a “scaled” Laplacian.

Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...

**2**

votes

**0**answers

225 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 ...

**1**

vote

**0**answers

22 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 ...

**1**

vote

**0**answers

78 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 ...

**1**

vote

**0**answers

100 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

58 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

47 views

### Harmonic functions subject to orthogonality condition

Suppose $(\bar{\Omega},g)$ is a Riemannian manifold with the metric being smooth where $\bar\Omega \subset \mathbb{R^3}$ is a compact domain with smooth boundary. Let $\Omega \subset \bar\Omega$.
...

**1**

vote

**0**answers

56 views

### Method of proving the regularity of the minimizer of geometric variational problems

Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer.
We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants ...

**1**

vote

**0**answers

42 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

56 views

### Regularity of Schrödinger Resolvent

The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...

**1**

vote

**0**answers

136 views

### Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let ...

**1**

vote

**0**answers

143 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...

**1**

vote

**0**answers

23 views

### Differential operator with codimension 2 singularity in the domain

The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ...