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

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4
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850 views

A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions. First recall the Cauchy integral formula: Let $U$ be an open subset of ...
4
votes
0answers
139 views

Uniqueness for solution of a d-dbar system related to Davey-Stewartson Solitons

This question concerns a system of equations that arise in the study of one-soliton solutions to the Davey-Stewartson equation. In what follows, $f(z)$ denotes a function which depends smoothly (but ...
4
votes
0answers
109 views

Regularity of reflection coefficients (or more generally the scattering transform)

Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$. We define the ...
3
votes
0answers
24 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
3
votes
0answers
54 views

Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space? More specifically given a smooth manifold of $M$ and a ...
3
votes
0answers
74 views

Regularity in PDE theory

I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not? Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...
3
votes
0answers
109 views

On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation $$\triangle \varphi = 4 \pi \rho \qquad (1)$$ defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and ...
3
votes
0answers
140 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in ...
3
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0answers
125 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi ...
3
votes
0answers
86 views

Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
3
votes
0answers
85 views

Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?

Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of $$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$ where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...
3
votes
0answers
138 views

Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$. Then we know that the eigenvalues of $-\Delta$ form an ...
3
votes
0answers
90 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
3
votes
0answers
119 views

$\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
3
votes
0answers
78 views

Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
3
votes
0answers
171 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
3
votes
0answers
109 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
0answers
92 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 ...
3
votes
0answers
86 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 ...
3
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0answers
252 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 ...
3
votes
0answers
168 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 ...
3
votes
0answers
106 views

Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation $- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin ...
3
votes
0answers
145 views

How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded

With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider $$u' + Au = f$$ in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
3
votes
0answers
192 views

Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...
3
votes
0answers
261 views

well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
3
votes
0answers
181 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
0answers
494 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 ...
3
votes
0answers
380 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D ...
3
votes
0answers
258 views

Best Poincare constants on the surface of a ball

I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first ...
3
votes
0answers
181 views

Numerical solution

Last time, I asked this question but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...
3
votes
0answers
371 views

kernel of the conformal Laplacian

Let $M$ be a smooth, closed manifold of dimension $n>2$. Let $L_g$ be the conformal Laplacian of the metric $g$. That is, $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar ...
3
votes
0answers
374 views

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases). Let $\nabla^{2}$ be the vector Laplacian. Let ...
3
votes
0answers
354 views

problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution? $$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...
3
votes
0answers
254 views

Controlling the Second Eigenvalue of a Schrödinger Operator

Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$. Let $L$ be the operator $$ L=\Delta+V $$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
3
votes
0answers
165 views

Generalizations of group algebras for arbitrary manifolds?

In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
2
votes
0answers
78 views

Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...
2
votes
0answers
38 views

Hess-Schrader-Uhlenbrock inequality for non-symmetric operators

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some ...
2
votes
0answers
74 views

Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël, http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf. The equations are $$ i\partial_tu+\Delta u=-|u|^{p-1}u $$ on ...
2
votes
0answers
99 views

Intuition behind Stokes operator?

I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is  $$A = -P_L Δ$$ where $Δ$ is the Laplacian, and $P_L$ is the Leray ...
2
votes
0answers
69 views

Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve $$\int_0^\infty\int_\Omega \nabla v\nabla ...
2
votes
0answers
103 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
2
votes
0answers
68 views

How analyze the following fully nonlinear equation

Now I want to consider the following pde $u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset ...
2
votes
0answers
88 views

Uniqueness of solution of elliptic equation with exponential nonlinearity

Consider the following equation $$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$ where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...
2
votes
0answers
64 views

Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful. I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...
2
votes
0answers
71 views

Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...
2
votes
0answers
82 views

long time existence of a nonlinear parabolic equation

I am thinking about a geometric problem which boils down to the following parabolic equation: Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$ $$\begin{cases}\displaystyle \frac{\partial ...
2
votes
0answers
217 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
2
votes
0answers
74 views

Is logarithmic convexity of the heat kernel with complex time a general fact?

Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
2
votes
0answers
147 views

Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$. I am ...
2
votes
0answers
111 views

Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem $$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...