**1**

vote

**1**answer

839 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**3**

votes

**0**answers

102 views

### What's a good resource for Hormander symbols of type (1/2, 1/2)?

I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...

**1**

vote

**0**answers

82 views

### Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
...

**3**

votes

**0**answers

89 views

### $L^2$ bounds for the gradient of subsolutions to parabolic equation

Suppose we have the differential inequality
$$
|\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|)
$$
in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...

**1**

vote

**1**answer

426 views

### solving elliptic system of first-order linear PDE's

I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which ...

**3**

votes

**1**answer

295 views

### Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...

**6**

votes

**1**answer

355 views

### Influence of Yau's solution to the Calabi Conjecture on the field of PDEs

I remember reading a long time ago(I can't recall where, unfortunately) that Yau's solution of the Calabi-Yau conjecture introduced new techniques that were very important for the field of partial ...

**-2**

votes

**1**answer

325 views

### A Poincare inequality for the Laplace-Beltrami operator [closed]

Suppose $w \in C^2 (S^{n-1}), \Lambda$ is Laplace-Beltrami operator on the sphere $S^{n-1}$, How can I prove follow Poincare inequality :
$\int_{S^{n-1}} w\Lambda w d\sigma \leq (1-n) \int_{S^{n-1}} ...

**1**

vote

**0**answers

80 views

### Analyticity of one-dimensional PDE solutions with respect to the space variable

Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients
$$
u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0,
$$
in some neighborhood of a point ...

**4**

votes

**0**answers

215 views

### Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...

**3**

votes

**1**answer

348 views

### traces of sobolev spaces under additional assumptions

Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$.
Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...

**2**

votes

**2**answers

209 views

### A general inequality about spherical mean of a function

suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...

**5**

votes

**1**answer

161 views

### On a conjecture of Lions for the wave equation

In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :
Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider
$\phi'' - \triangle \phi = F$
...

**8**

votes

**0**answers

180 views

### Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the ...

**0**

votes

**0**answers

283 views

### Analytic solution of Poisson's equation

Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.
Impose a von Neumann boundary condition ...

**2**

votes

**1**answer

218 views

### Reference request: harnack inequality for distributional solutions of the heat equation

Dear Math Overflowers,
I'm looking for references on the parabolic Harnack inequality for distributional solutions of the heat equation on the whole space
$$
\partial_t u=\Delta u\quad\text{and}\quad ...

**0**

votes

**1**answer

743 views

### Existence of Green's function and the Dirichlet problem

Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, ...

**31**

votes

**1**answer

2k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**1**

vote

**0**answers

192 views

### Focusing NLS: $L^2$ convergence of a solution as $t\rightarrow +\infty$

Consider the cubic focusing non linear Schrodinger equation in dimension $n\geq 2$:
$$(iu_t+\Delta)=-|u|^2u\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}^n)$$
Can we find an initial data $u_0\neq 0$ such ...

**1**

vote

**1**answer

133 views

### Image of the trace operator on W^{1,1}

Let $\Omega \subset R^n$ be a bounded region with Lipschitz boundary. Is the trace operator $T: W^{1,1}(\Omega)\rightarrow L^1(\partial \Omega)$ surjective? If not, what is the image?

**1**

vote

**3**answers

274 views

### What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is ...

**2**

votes

**2**answers

187 views

### Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...

**0**

votes

**0**answers

153 views

### Can a function be constructed from the direction of its gradient?

Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...

**1**

vote

**0**answers

116 views

### localization of the $L^p$ variation for heat equation

I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is.
The ...

**1**

vote

**1**answer

175 views

### Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...

**1**

vote

**1**answer

277 views

### Sobolev Inequality

Let $\Omega$ be a bounded region in $R^n$ and define
$W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$
where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that
...

**1**

vote

**0**answers

137 views

### Extension of solutions of PDE

Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let ...

**11**

votes

**2**answers

506 views

### Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...

**1**

vote

**2**answers

321 views

### How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$.
More ...

**1**

vote

**0**answers

133 views

### How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...

**0**

votes

**1**answer

149 views

### weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions

I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...

**3**

votes

**0**answers

264 views

### is this a known method for solving PDEs

I recently posed a system of PDEs to solve on MSE at http://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions.
However, in this post, I'd like to show here ...

**1**

vote

**1**answer

120 views

### Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$

Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that ...

**2**

votes

**1**answer

187 views

### functions of bounded variation and gradient vector measure

I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} ...

**1**

vote

**1**answer

139 views

### A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) ...

**8**

votes

**3**answers

325 views

### Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...

**0**

votes

**1**answer

197 views

### carleman inequality

Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE ...

**1**

vote

**2**answers

213 views

### vector valued pde's good reference

I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows.
Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n ...

**1**

vote

**0**answers

46 views

### parabolic PDE with pseudomonotone operators

I am looking for a reference where well-posedness of problems
$$u_t + A(t)u = f$$
is addressed via the Galerkin method where $A$ is a pseudomonotone operator. I am aware that Roubicek's book ...

**5**

votes

**1**answer

342 views

### Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO ...

**18**

votes

**4**answers

960 views

### When to use more exciting function spaces than ordinary Sobolev spaces?

In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces ...

**0**

votes

**0**answers

180 views

### If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?

Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in ...

**2**

votes

**1**answer

199 views

### Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ...

**3**

votes

**0**answers

189 views

### W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...

**2**

votes

**1**answer

278 views

### Fully non-linear PDE

A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...

**1**

vote

**1**answer

136 views

### radial limits of subharmonic functions

Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every ...

**2**

votes

**0**answers

108 views

### Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...

**2**

votes

**0**answers

168 views

### A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE
$$\frac{d}{dt}D^0_t(\cdot) = ...

**1**

vote

**1**answer

237 views

### Blow up of solutions to parabolic PDEs

I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice ...

**2**

votes

**2**answers

151 views

### Changing the space of test functions in PDEs

Let $V \subset H \subset V'$ be a Hilbert triple.
We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T ...