**1**

vote

**0**answers

54 views

### Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...

**0**

votes

**0**answers

69 views

### Compact Embeddings [on hold]

Put:
$D=\{u\in L^{2}(\mathbb{R}^{2})| N=(x\frac{d}{dy}- y \frac{d}{dx})u\in L^{2}(\mathbb{R}^{2}) \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n})$ with compact injection?
Thank you in advance.

**3**

votes

**0**answers

45 views

### A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0
$$
Here $a_i:...

**0**

votes

**0**answers

60 views

### why $\varphi''\in L^{2}(R)$ [closed]

I have the following question: Let $T_{c}$ be an unbounded operator with domain $D(T_{c})=\{u\in L^{2}(R), T_{c}(u)\in L^{2}(R)\}$.
If $\forall \varphi \in \mathcal{C}^{\infty}_{0}(R): \|\varphi''\...

**0**

votes

**0**answers

19 views

### Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy,
$$
\int_{\Omega}|\nabla u |^2\, -\, \int_{\...

**1**

vote

**0**answers

48 views

### Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator:
$ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$,
where
$N=(x \frac{d }{dy} -y \frac{d }{dx})$ ,
...

**1**

vote

**0**answers

44 views

### Mean Value Property for harmonic functions [migrated]

I would appreciate any insights on this matter;
Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...

**5**

votes

**1**answer

198 views

### finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...

**0**

votes

**0**answers

64 views

### Solving a system of Laplace equations

Let $u_0$ and $u_1$ to be smooth functions defined on $\Omega\subset\mathbb{R}^n$, consider the following system of equations
$$\triangle u_1 = C_1(\partial_{ij}u_0),$$
$$\triangle u_0 = C_0u_1,$$
...

**5**

votes

**1**answer

190 views

### minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...

**0**

votes

**1**answer

123 views

### a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(R^{2})$?
Thank you in advance.

**0**

votes

**0**answers

41 views

### When can an analytical solution for the heat equation be obtained?

I am currently trying to model a system with a time varying heat flux. It seems most researchers are using FEM to obtain the heat distribution (solve the heat equation).
When can the heat equation be ...

**2**

votes

**1**answer

99 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...

**0**

votes

**0**answers

21 views

### how can we extend this result [duplicate]

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with
$D$ contains a Schawrz space $S$
$\Big<\psi,T_{a}\varphi\...

**1**

vote

**0**answers

84 views

### Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.
The most ...

**1**

vote

**0**answers

70 views

### Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....

**0**

votes

**0**answers

60 views

### Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on $\...

**3**

votes

**0**answers

59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...

**0**

votes

**0**answers

25 views

### Holder regularity of a singular elliptic problem

In the view of the paper https://pdfs.semanticscholar.org/063b/59c1f070c9b58ab88ed7ef76209187ce7e5c.pdf,
I have a problem about Theorem 1.1 (iv) , I do not understand how $I \leq cd^{1-\alpha}(x_1,...

**3**

votes

**1**answer

136 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...

**1**

vote

**2**answers

67 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

**1**

vote

**0**answers

93 views

### Weyl's law for minimal surfaces

I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...

**7**

votes

**1**answer

112 views

### Properties of connection Laplacian on vector fields

Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...

**0**

votes

**0**answers

81 views

### Continuity of solutions of nonlinear elliptic PDEs

Consider the nonlinear 2nd order elliptic PDE $$\sum_{i, j} a_{ij}(x, t) \partial_i\partial_j u + \sum_k b_k(x, t) \partial_k u + c u = F(u), \quad x \in \mathbb{R}^n, t \in [0, \infty).$$
Here $a_{ij}...

**1**

vote

**0**answers

27 views

### Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...

**0**

votes

**0**answers

60 views

### Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\...

**1**

vote

**1**answer

62 views

### leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...

**2**

votes

**2**answers

147 views

### Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...

**8**

votes

**0**answers

170 views

### Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...

**3**

votes

**0**answers

44 views

### Interior regularity for elliptic operators with non smooth coefficients

I need a pretty standard interior regularity result for a second order elliptic operator of the form
$$
-\nabla^b \cdot (A(x) \nabla^b v)+c v=f,
\qquad
\nabla^b=\nabla+ib(x)
$$
where $A(x)$ is a ...

**2**

votes

**0**answers

51 views

### Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$.
My question concerns solutions to $\triangle_g u =0$ that are say ...

**1**

vote

**0**answers

69 views

### An H2 estimate for Helmholtz equation

How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...

**1**

vote

**0**answers

36 views

### About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...

**2**

votes

**1**answer

261 views

### Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...

**1**

vote

**1**answer

125 views

### Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...

**1**

vote

**1**answer

64 views

### Why are the tangential derivatives in this diffraction problem zero? [closed]

I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in $\...

**2**

votes

**0**answers

122 views

### The minimum value of a energy integral

Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0
$ and
$${\nabla ^2}\...

**0**

votes

**1**answer

84 views

### Explicit solution for one-dimensional Gelfand problem

I wonder if the ODE
$y''+e^{y}=a$
can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions
$y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$...

**3**

votes

**0**answers

40 views

### Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold :
$ \| e^{\tau \phi} \triangle u \|_{L_{\delta}^2({\mathbb{R^3})}}> C \tau \| e^{\tau \phi} u \|_{L^2_{\...

**2**

votes

**1**answer

58 views

### What is the function space $H^1_{m, \sigma}$?

I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order".
Theorem 3.1 is the first time in their paper that the function space $...

**6**

votes

**2**answers

206 views

### Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (http://...

**0**

votes

**0**answers

56 views

### Bound of solution of pde with a distance function

Hello，i am going to solve the PDE,
$\Delta u = -K in \Omega $and u=0 on the boundary ，where K is some poistive constant.
Then i have read a paper which stated that u>=distance(x，boundary of $\Omega$...

**2**

votes

**0**answers

114 views

### How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on $\...

**0**

votes

**0**answers

22 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...

**2**

votes

**2**answers

202 views

### Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function (...

**2**

votes

**1**answer

154 views

### Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...

**1**

vote

**1**answer

103 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 $u$ ...

**2**

votes

**0**answers

101 views

### Dirichlet-to-Neumann Map is selfadjoint

Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$.
For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...

**3**

votes

**0**answers

104 views

### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...

**2**

votes

**0**answers

115 views

### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...