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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2
votes
1answer
81 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
2
votes
1answer
93 views

On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite ...
2
votes
0answers
62 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
0
votes
2answers
126 views

A basic question about stability of level sets

Let $u_i \in C^1(\Omega)$ with $|\nabla u_i|>0$ in a simply connected region $\Omega$ with connected boundary, and $u_1=u_2$ on $\partial \Omega$. Assume $$ \nabla u_i(x) \cdot V_i (x)=|\nabla u_i(...
2
votes
0answers
69 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
3
votes
2answers
176 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
0
votes
0answers
29 views

Moser inequality involving the trace operator

Can anyone give me a reference of whether there exists a Moser-Trudinger type inequality involving the trace operator. More precisely, $$ \int_{a}^{b} \exp\bigg[ \beta U(x, 0)^2\bigg]\leq C$$ for ...
9
votes
1answer
193 views
+50

A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. Assume $d \ge 3$ ...
8
votes
2answers
263 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
3
votes
0answers
93 views

gaussian upper bound on spherical heat kernel

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?
2
votes
1answer
62 views

Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
3
votes
0answers
71 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal ...
3
votes
0answers
51 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
7
votes
1answer
178 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
1
vote
0answers
54 views

Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?

Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set. Is it true that $(S,w)$ is metrizable? Since $S$ is relatively weakly compact, it ...
11
votes
1answer
369 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
0
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0answers
47 views

Sobolev embeddings for vector-valued functions

I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space. In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
2
votes
1answer
141 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
1
vote
0answers
93 views

Support of a microlocal defect measure

I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
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vote
0answers
31 views

Geometric control condition

In control theory of hyperbolic equations, we talk always about the G.C.C(Geometric control condition), it is a constraint on the domain to ensure the exact controllability of hyperbolic systems but I ...
1
vote
1answer
120 views

Applications of the Calderon-Zygmund theory to PDE's

I am planning to build a PDE topics course focussing on the Calderon-Zygmund theory. I know some important applications of the Calderon-Zygmund theory to elliptic PDEs, but I don't know enough to get ...
2
votes
0answers
69 views

For what functions does Nash inequality becomes equality?

For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)? Also same question about Poincare inequality.
2
votes
1answer
134 views

*Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.) Consider the one-dimensional heat equation $$\...
3
votes
2answers
170 views

Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$

Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$ with initial data $u(0,\cdot) = u_0$. Fix $\alpha >0$. How can I estimate (without using explicitly ...
0
votes
0answers
59 views

Local and global analysis of PDEs [closed]

The general theory of linear partial differential equations is largely addressed to local problems, i.e.,to the study of solutions in a suitably small neighbourhood of $x_0\in \mathbb{R}^n$. It is ...
0
votes
0answers
29 views

von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here ...
2
votes
0answers
41 views

Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$. In the ...
2
votes
0answers
172 views

Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$ where $x_0$ is an arbitrary but fixed ...
1
vote
2answers
65 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
3
votes
1answer
230 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
2
votes
3answers
241 views

Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation. I encountered the following integral equation for functions $f:[...
1
vote
1answer
84 views

Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$u(t) \leq \psi(t) ...
1
vote
1answer
70 views

Method of characteristics for 2x2 systems

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation, but it is not so easy to do the same for PDE systems. Let's say we have ...
1
vote
0answers
82 views

How can I obtain this inequality (from Evan's PDEs)

I am self studying PDEs from Evans' "Partial Differential Equations" textbook. Currently, I am going through Theorem 1 from Section 5.7 (Rellich-Kondrachov compactness theorem) and am having ...
1
vote
0answers
55 views

Relative boundedness of the adjoint

Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$ ...
2
votes
0answers
148 views

A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$. Now define the operator $ \mathcal{A} : C^{‎\sigma‎, \sigma‎/2‎}(‎X‎) \to C^{‎\sigma‎, \...
0
votes
1answer
156 views

Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
1
vote
0answers
75 views

2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of $$ u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\ u(0) = u_0 \text{ in } H, u'(0)...
0
votes
0answers
29 views

What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?

Let the generator $L$ of a diffusion process be given in Hörmander form, i.e. $$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$ where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...
1
vote
0answers
58 views

Propagation of singularities and the Schrodinger equation

I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation $$(i \partial_t-p(x,D))...
1
vote
1answer
80 views

Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
2
votes
1answer
97 views

Support of functions in Fourier domain

Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...
5
votes
1answer
104 views

Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$ The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
1
vote
1answer
122 views

$L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$ Such a function has the property that when multiplied with any ...
1
vote
1answer
154 views

Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
10
votes
2answers
347 views

$x f'$ bounded by $x^2f $ and $f''$?

Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$ I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
0
votes
2answers
114 views

Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
2
votes
1answer
46 views

order of the singularity of a Green's function to the fractional Laplacian

I was looking at a problem which involves the Green's function of a fractional Poisson equation. To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and \begin{equation} \begin{...
0
votes
0answers
22 views

Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation \begin{array}{c} y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\ y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\ u(...
7
votes
2answers
161 views

Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...