Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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2
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1answer
78 views

Conditions for existence of $m$-th differentiable root of a non-negative definite matrix

In M.I. Friedlin's famous paper "On the Factorization of Non-Negative Definite Matrices", he shows that if a non-negative definite symmetric matrix $a(x)=\{a^{ij}(x)\}_{i,j=1}^n$ is in ...
4
votes
1answer
144 views

K-Theory of Algebra of Zeroth Order Pseudo differential operators

Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators? Thanx!
1
vote
0answers
83 views

Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional

In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system: \begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, ...
0
votes
2answers
145 views

Interior Schauder estimates with weights

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a ...
2
votes
1answer
111 views

Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
2
votes
0answers
66 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
1
vote
1answer
131 views

If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function. When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
2
votes
1answer
124 views

The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x } $ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
1
vote
1answer
106 views

About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps: If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$: ...
2
votes
0answers
80 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 ...
3
votes
1answer
284 views

About Aubin-Lions Lemma

I have a question about Aubin-Lions Lemma, the standard Aubin-Lions lemma need those Banach Space be reflexive spaces, are there any version of Aubin-Lions without reflexivity? Standard ...
0
votes
2answers
123 views

property of local sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
2
votes
0answers
70 views

Linear heat equation with initial condition of generalized function

I am consider a very simple heat equation over the interval $[0, 1]$ with a Neumann BC and a very bad initial condition, written as: $\partial_tu(t, x) = \partial^2_xu(t, x) + a(t, x)u(t, x)$, for ...
2
votes
1answer
103 views

An interpolation type inequality

Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$ $$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le ...
2
votes
2answers
158 views

Sobolev spaces on boundaries

Consider the Sobolev space $W^{s,2}=H^s$ for $s=\frac{1}{2}.$ Let $\Omega \subset \mathbb{R}^n$ be an open set with boundary $\partial\Omega$. I have seen two definitions of the space ...
1
vote
1answer
112 views

If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?

Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$ where $w'$ means the weak ...
3
votes
3answers
201 views

Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces?

For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way. Define ...
4
votes
0answers
256 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 ...
1
vote
0answers
107 views

Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations singular at some point? For example, I am analyzing a partial differential equation $$ ...
4
votes
1answer
207 views

What's wrong with the Courant nodal domain theorem

The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. ...
5
votes
2answers
255 views

If a PDE have a unique classical solution, must it have a unique viscosity solution?

If a PDE have a unique classical solution, must it have a unique viscosity solution? The particular problem I am interested in is parabolic, but I would be interested in the general case. A short ...
1
vote
1answer
110 views

Differences between parabolic operators of second order and higher order

Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly ...
1
vote
0answers
42 views

Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying: \begin{align} \Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...
17
votes
1answer
712 views

The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206). If ...
3
votes
0answers
166 views

Reference on a Monge-Ampère-like equation

We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$: $$ \det(Hess(w))=1~, $$ ...
4
votes
2answers
156 views

First order Elliptic operator

Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$? For example, is the dimension of $V$ ...
2
votes
0answers
70 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
2answers
111 views

Maximum of the solution of a parabolic PDE

Let $u:\mathbb{R}\times [0, \infty) \rightarrow \mathbb{R}$ be defined by $u_{xx} + u_x - u_t = u(u - 2)(u - 1)$ with $u \rightarrow 0$ as $|x| \rightarrow \infty$ and $u(x,0) = 3e^{-x^2}$. Now, let ...
3
votes
0answers
70 views

Subordination identity and heat operator

I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...
2
votes
1answer
155 views

Proof of regularity for bounded elliptic problem

We consider the boundary value problem for potential in the form: $$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$ with boundary conditions $$\nabla ...
1
vote
1answer
348 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 $(\Omega,\Sigma,\mu)$ ...
2
votes
0answers
57 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
0answers
57 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
0answers
79 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 ...
0
votes
1answer
194 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 ...
2
votes
1answer
195 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 $0$ ...
5
votes
1answer
284 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
1answer
145 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
0answers
68 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 ...
2
votes
0answers
175 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 ...
2
votes
1answer
238 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 ...
1
vote
2answers
166 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
1answer
125 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 $\phi'' - \triangle \phi = F$ ...
7
votes
0answers
136 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
0answers
62 views

$|\nabla f|^2 \in W^{1,2}(M)$ for harmonic function f on manifolds 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. ...
0
votes
0answers
220 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
1answer
162 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
1answer
305 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, ...
0
votes
0answers
133 views

Cheeger-Gromov-Taylor theory on manifolds with boundary

I was reading the paper "Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds" by Cheeger, Gromov and Taylor and I am ...
17
votes
0answers
408 views

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

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it. Recently I encountered in a class the fact that there is a generating function of ...