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

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An Estimation on the Heat Kernel

I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get ...
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64 views

A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...
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38 views

A $\mathcal{C}^1$ differentiable domain is $F_\sigma$?

Let us consider a domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that for every $R>0$ the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. ...
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1answer
184 views

Is Lax-Milgram true without the separability assumption?

I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam: Let $X$ be a separable Hilbert space (norm $\|\cdot\|_X$) and let $$ a:X\times X\to\Bbb{R} $$ be a bilinear continuous ...
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1answer
112 views

$H_0^1(\Omega)$ in the study of the Navier-Stokes Equations

This is cross-posted on MSE: http://math.stackexchange.com/q/1584519/9464 Let $\mathcal{V}$ be the space (without topology) $$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot ...
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1answer
229 views

Trying to solve a linear PDE… I thought it was simple

I have a PDE of the following form, from a physics problem: $$ y \left(\alpha \frac{\partial }{\partial y}+x \frac{\partial^2 }{\partial x \partial y} \right)f(x,y) = \left( z_1 + z_2 x^\alpha y^{-2} ...
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103 views

A priori estimate for diffraction problem for linear elliptic PDEs

I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation. I looked at ...
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86 views

Commutator representation of certain smoothing operators

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
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193 views

Compact Eucledean hypersurfaces with “almost” constant H_k curvature

Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...
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1answer
178 views

Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes: Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
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139 views

On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $

This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data: $ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) ...
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1answer
75 views

Boundary behaviour of a second order pde with characteristics

Good morning everybody. My question is inspired from the following fact: Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial_x^2 ...
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49 views

Regularity result for Neumann problem

I have two questions. On Elliptic regularity for the Neumann problem, the OP asked whether the test function $v$ must be of mean value zero. However, isn't it true that we only need $f$ is of mean ...
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60 views

Well-posedness of gradient flows

For a convex lower-semicontinuous functional on a Hilbert space $I\colon H\rightarrow\mathbb{R}$, it is shown in Evans' PDE that the Hilbert-space-valued ODE $$\begin{cases}\mathbf{u}'(t)\in-\partial ...
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2answers
250 views

How to prove Liouville measure is invariant under geodesic flow?

Let $M$ be a complete n dimensional Riemannian manifold. $vol$ denotes the n dimensional Hausdorff measure. Let $$ SM=\{(x,v)|x\in M, v\in T_xM, \|v\|=1\} $$ be the unit tangent bundle of $M$. Then ...
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143 views

A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy $-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$, where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
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39 views

Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem ...
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382 views

A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in ...
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1answer
127 views

Linear Schrödinger equation on $\mathbb{H}^{d}$

Consider the linear Schrödinger equation $i\partial_t u = -\Delta u$, where $\Delta$ is the Laplacian on the hyperbolic space $\mathbb{H}^d$. What are the admissible pairs $(p, q)$ such that we have ...
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56 views

Solvability Helmholtz equation [closed]

Let $D=D_1(0)\subset\mathbb{R}^2$ and $\lambda\in\mathbb{R}$, $\lambda>0$. Consider the Helmholtz operator $L=(\Delta +\lambda I).$ Let $f\in Ker_0(L)$, that is $f$ solves $$ Lf=0\quad\text{ in ...
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49 views

method for global existance for the NLS

We consider the nonlinar Schr\"odinger equation(NLS): $$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$ where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...
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1answer
140 views

separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
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2answers
118 views

Positivity of semiclassical pseudodifferential operators

Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis) Let $m\in\mathbb{R}$, and ...
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70 views

Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
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1answer
104 views

Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$

Is there anything known about uniqueness of classical solutions to $$ \partial_t u -u\Delta u=0\quad u(0,\cdot)=1 $$ on smooth domains $[0,T]\times D$ without boundary conditions? I know that ...
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1answer
100 views

How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results: Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in ...
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61 views

Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity). Let $ \Omega$ denote a cube in $ R^n$ and consider ...
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51 views

Error in the paper “Oscillating-Decaying Solutions, Runge Approximation and its Applications to Inverse Problems”?

I am going to use simpler notation then the paper. The potential problem is not very technical. Theorem 4.2. in the paper states the following: a) If $t < t_0$, then $I(t,\tau) \to 0$ as $\tau ...
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74 views

Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :) Consider the following Laplace boundary value problem (BVP) $$\matrix{ {{\nabla ^2}\Phi ...
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108 views

A construction of the fundamental solution for Schroedinger equations

Does someone know some book or lecture notes useful for the reading of the paper "A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 ...
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69 views

Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde $$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...
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95 views

A modification of Minty's trick?

I have the following result: $$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$ where $a$ and $b$ are both non-negative. Does it follow that $b(t) = g^{-1}(a(t))$? ...
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1answer
180 views

Progress on isospectral plane domains

Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit ...
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1answer
199 views

Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...
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1answer
74 views

Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
4
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0answers
354 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
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2answers
664 views

Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
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26 views

Regularity of a flux induced by a potential

Take $\Omega\subset R^n$ with smooth boundary (take a ball for example) a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$ a ...
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0answers
74 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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1answer
88 views

The monotone operator in $BV$ space

I am considering the following minimizing problem: $$ \min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\} $$ where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
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1answer
84 views

Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE

Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for ...
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38 views

Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
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46 views

The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question... Let $\Omega\subset \mathbb R^2$ be open ...
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65 views

PDE Parameter-Dependent Center Manifolds

In the book Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems by Mariana Haragus, parameter-dependent center manifolds are discussed. Here it is assumed ...
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5answers
511 views

List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
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1answer
78 views

Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says: Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$. ...
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1answer
208 views

stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)? Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases} ...
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1answer
110 views

Elliptic regularity with mixed boundary conditions

I'm looking for some results about elliptic regularity with mixed boundary conditions. I know they exist with non mixed boundary conditions but where can I find some results for the mixed case? Thanks ...
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101 views

Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations: $(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$ $(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$ for ...
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174 views

Boundary conditions in the Finite Element Method

I just want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} (p(x)u'(x))'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do ...