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
votes
1answer
367 views

Hölder estimates for the Complex Monge-Ampere equation

If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in ...
2
votes
1answer
388 views

Distributional derivative is locally integrable - then the distribution as well?

Given a distribution $T \in D'(\mathbb{R})$ such that the distributional derivative $\partial T \in L^1_{loc}(\mathbb{R})$. Can one deduce that $T \in L^1_{loc}(\mathbb{R})$ as well? Or can anyone ...
1
vote
2answers
834 views

Geometric Mean Value Property

Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...
1
vote
0answers
113 views

Strichartz estimates over cones

I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality: ...
2
votes
1answer
289 views

A Sobolev-type inequality with weights

In the study of a particular PDE I found myself wanting to prove the following inequality: $( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r ...
5
votes
2answers
364 views

Willmore minimizers for genus $\geq 2$

For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as $$ \cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g, $$ where $\vec H$ is the mean ...
2
votes
2answers
422 views

Monge Ampere equations (concavity)

The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of ...
6
votes
3answers
538 views

Moser regularity proof avoiding John-Nirenberg lemma

I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely ...
2
votes
2answers
1k views

Weak maximum principle / comparison principle for parabolic equations with Neumann conditions

Hello everyone. I'm about to use a comparison principle that I belive is true, but I can't find any precise reference to be sure of it. Here is what I would like to say : I have a parabolic equation ...
7
votes
3answers
1k views

unique continuation principle

I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on ...
7
votes
2answers
629 views

Chebyshev net in 3D

I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic. This question boils down to the PDE described below. (I do not know much about PDEs, so feel free to say ...
2
votes
2answers
544 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
1
vote
1answer
440 views

Existence of cut-off functions in metric spaces

Let $X_1,\ldots,X_m$ be Lipchitz continuous vector fields in an open set $\Omega \subset \Re^n$, Let $d(\cdot, \cdot)$ denote the control distance associated to $X$. With respect to this control ...
2
votes
1answer
144 views

ellipticity independent of metric?

I am new to the theory of pseudo-differential operators on compact manifolds, but I need to use a result related to this theory in a proof I'm working on. The problem is as follows: Let $(M,g)$ be a ...
3
votes
2answers
878 views

What is soliton

I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking.. In literature, i am finding following words:(Wikipedia+ others). ...
7
votes
1answer
348 views

Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
4
votes
1answer
226 views

a question about Lp norm of curvature on convex curves

Suppose we have two strictly convex closed curves $C_{1}$ and $C_{2}$, $C_{1}$ contains $C_{2}$, then can we conclude $\int_{C_{1}} \kappa_{1}^{p} ds\geq \int_{C_{2}} \kappa_{2}^{p} ds$, $\kappa_{1}$ ...
2
votes
0answers
143 views

A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# ...
0
votes
0answers
85 views

A slightly subcritical elliptic equation on the ball; blow-up behavior near zero

I am interested in positive ground state solutions of the following elliptic pde: $-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ ...
2
votes
0answers
81 views

Asymptotics of quasilinear elliptic equations with Dirac right hand side

On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form $$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha ...
0
votes
1answer
393 views

Maximum principle for heat equation on infinite domain

Let $u(x, t)$ be a solution of $u_t=u_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in ...
1
vote
1answer
733 views

Solution of Heat equation with Neumann BC in an arbitrary domain

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary. Is this true: Any solution ...
2
votes
0answers
154 views

Integrability of ground state solution for elliptic equation

For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...
3
votes
1answer
247 views

ellipticity and invertible differential operators

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}_+$. Suppose that the differential equation $Pu=f$ has a unique ...
7
votes
2answers
495 views

Methods for determining domains of influence

Given a hyperbolic PDE, the domain of influence of a spacetime point $x$, say $I_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them geometric ($I_x^G$) and ...
1
vote
0answers
171 views

Weak solution of a certain pde with integral term

Let us consider the following pde on the domain $(0,T)\times(0,1)$ $ \dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0 $ with initial data $p(0,x)=p_{0}(x)$ and boundary data ...
1
vote
0answers
173 views

Is this Stefan-type problem an open problem?

I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
1
vote
1answer
134 views

What we mean by positive solution and radial solution of any partial differential equation

I am not getting the most of the article concern with the existence and uniqueness of positive solution and radial solution. I just want to know how a positive solution and radial solution of ...
18
votes
5answers
2k views

H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations ...
3
votes
2answers
591 views

Asymptotic expansions of solutions of nonlinear PDE's

Let's consider the following Cauchy problem: $$u_t=\alpha(x,t,u)u_{xx}+\beta(x,t,u)(u_x)^2+\gamma(x,t,u)u_x+\phi(x,t,u)$$ $$u(x,0)=u_0(x).$$ Assume that functions $\alpha,\beta,\gamma,\phi$ are ...
1
vote
1answer
310 views

Reparametrizing characteristic curves for PDE's

I'm looking for solutions for a PDE that looks like this $$ \nabla u(\vec x) \cdot f(\vec x) = k. $$ For some clarification, $u$ is a log-probability. And this arises from a Fokker-Plank-like ...
2
votes
0answers
257 views

Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$, \begin{eqnarray} Lu &=& \frac{\partial u}{\partial t}, u(x,0) &=& u_0 \; \; ...
5
votes
2answers
487 views

Blow-up for the quasilinear heat equation $u_t= u \ u_{x x}$ or the related $w_t= \left(w_x e^w\right)_x$

What kind of approaches can be used to study the following quasilinear parabolic pde for a scalar function $u=u(x,t)$ ? $$ u_t= u \ u_{x x} $$ The physical problem where this pde comes from dictates ...
2
votes
1answer
168 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
7
votes
2answers
692 views

Solving PDE via Cellular Automata

Is there a theory for solving PDE by using Cellular Automata ? Something which is on the line of, passing to the limit (scale) i.e., if you increase the number of grid points the solution to the ...
3
votes
0answers
174 views

Is this integral operator about Stokes' Flow compact?

Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]: $$ ({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 ...
3
votes
3answers
712 views

Reference request: Simple facts about vector-valued Sobolev space

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = ...
8
votes
2answers
496 views

The ground state is signed and symmetric

Background In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action ...
0
votes
1answer
234 views

Nonlinear PDE ${u_{tt}}^2u_{ttxx} = 1$

I have been trying to solve this equation during fortnight $$ {u_{tt}}^2u_{ttxx} = 1. $$ But I still here. The only thing is change of variables $u_{tt}(t,x) = y(t,x) $ and solved the ODE $y'' = ...
11
votes
4answers
744 views

Eliminating 1st order terms in elliptic partial differential equation

Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves the 2nd ...
4
votes
3answers
454 views

Is there a PDE for this phenomenon?

At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface. Is there a physical law - like the heat equation - that describes the flow? Will ...
7
votes
2answers
916 views

Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold

Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...
6
votes
0answers
269 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...
0
votes
2answers
478 views

Flow of evolutionary vector fields

Consider a smooth vector bundle $\pi: E\rightarrow M$, the associated infinite jet bundle $J^\infty(\pi)$, and evolutionary vector fields $\partial_\varphi = ...
0
votes
1answer
521 views

Hopf Boundary Point Lemma

The Hopf Boundary Point Lemma http://en.wikipedia.org/wiki/Hopf_lemma is a result for the unit normal vector field and the normal derivative. Is it true if one considers arbitrary directional ...
1
vote
2answers
164 views

vector valued BVP for ODE's

I am dealing with a vector valued second order homogeneous BVP: $\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$ where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and ...
8
votes
1answer
337 views

When the adjoint of a hypoelliptic operator hypoelliptic

Assume, $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$. Recall that $L$ is a hypoelliptic differential operator, if for ...
2
votes
0answers
180 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
0answers
452 views

Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
0
votes
1answer
144 views

Brezis-Nirenberg result compared to abstract bifurcation theory

Dear Mathoverflow'ers, I am interested in the following equation: $-\Delta u = u^{p-1} + \lambda u$ in $ \Omega$ with $ u=0 $ on $ \partial \Omega$. 1) My question is related to the ...