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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91 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
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88 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 ...
2
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100 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
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159 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) = ...
2
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94 views

Elliptic equations with divergence-free drift terms

Given $\ \mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ bounded, $div$$(\mathbf{u})=0$, ...
2
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85 views

Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in ...
2
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127 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
2
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66 views

Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
2
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119 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
2
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114 views

linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator: $$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$ where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
2
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94 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
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380 views

Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real ...
2
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109 views

Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
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106 views

Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
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0answers
167 views

Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation $$u_t-iu u_x-iu_{H,xx}=0$$ where $u_H(x,t)$ denotes the ...
2
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83 views

P-laplacian equation

Hi guys, in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ? Thank you!
2
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160 views

Extension divergence-free, curl-converging vector field

Hi. Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
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198 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
2
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63 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question. Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose, ...
2
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103 views

Best constant of Gagliardo-Nirenberg inequality in exterier domain

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla ...
2
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229 views

Constant in the Poincare inequality for curl square integrable vector fields

$\newcommand{\v}[1]{\boldsymbol{#1}}$For an $u\in H^1(\Omega) = W^{1,2}(\Omega)$ where $\Omega$ is Lipschitz, we have $$ {\|u - \frac{1}{\Omega} \int_{\Omega} u\|}_{L^2}\leq C {\|\nabla u \|} $$ ...
2
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322 views

How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...
2
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215 views

Recovering full regularity by energy method in the heat equation

Consider the heat equation $$ u_t = u_{xx} + f, $$ on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
2
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144 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/# ...
2
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0answers
82 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 ...
2
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157 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 ...
2
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280 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 \; \; ...
2
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182 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
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238 views

Equation of motion of a charged string in a twisted torus.

The flat torus background: Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...
2
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481 views

Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background Consider the one dimensional second order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} ...
2
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0answers
148 views

Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense. Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...
2
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0answers
274 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
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265 views

General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that ...
2
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236 views

Can the solution manifold for an exterior differential system be represented using alternating multivectors?

Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For ...
2
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0answers
292 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
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0answers
280 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
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222 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 ...
2
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0answers
218 views

Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary ...
1
vote
0answers
10 views

Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse. Does there ...
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73 views

Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...
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0answers
28 views

ABP estimates for semiconvex functions

Referring to the classical ABP Estimate (Gilbarg-Trudinger Lemma 9.2) I am looking for if such an estimate can be generalized to semiconvex functions. In an article of Trudinger (Comparison Principles ...
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31 views

Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)? I am thinking of eg. ...
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0answers
21 views

Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?

Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...
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0answers
42 views

decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
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0answers
59 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
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107 views

How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

[This may not be a research-level question; if it violates any term of this website, I will delete it right away] My question is quite simple: Suppose we are given a PDE of with a boundary condition ...
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0answers
53 views

Bogomol’nyi’s Formula for the Critical Action

I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is ...
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0answers
118 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 ...
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0answers
47 views

Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$. Consider the system, with $u^\epsilon, v^\epsilon ...
1
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0answers
91 views

A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...