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

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The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
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1answer
86 views

Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e. $$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...
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1answer
109 views

Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
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83 views

Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold. If it is necessary, we could consider the $b_1(Y)=0$ case. Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
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1answer
194 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
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39 views

$L^\infty$-contractive semigroups

Let $L^\infty(\mathbb T)$ be the space of $2\pi$-periodic and bounded measurable functions and $\mathcal P$ be a pseudo-differential operator defined on $\mathcal D(\mathcal P)\subset L^\infty(\...
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28 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
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0answers
21 views

About Cahn - Hilliard equation solution uniqueness

The uniqueness of the solutions of the Cahn - Hilliard nonlinear PDE $$\dfrac{\partial c}{\partial t}=\nabla\dot{}(M\nabla\mu)$$ has been proved for many form of the chemical potential $\mu$. What ...
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114 views

Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) lead to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose ...
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1answer
59 views

Domain of the Stokes operator

Let $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$) $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$ $\mathcal H:=\overline{\mathfrak ...
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2answers
762 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 ...
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41 views

Questions about the regularity of the solution of the heat equation in a bounded domain [closed]

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
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92 views

The eigenfunction of modified $1$-laplace equation?

Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation $$ -\Delta u=0 $$ has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
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1answer
111 views

Global Poincaré type estimate

For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
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3answers
162 views

Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me. Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...
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1answer
74 views

The dependence of constant in a trace theorem on the diameter of domain

The trace theorem says for a nice domain, say bounded Lipchitz domain, $\Omega$, we have $$\left\Vert Tu \right\Vert_{H^{1/2}(\partial \Omega)} \leq C \left\Vert u \right\Vert_{H^1(\Omega)},$$ where $...
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0answers
21 views

Existence of solution to first order pde [closed]

Let $U : = \big(-\frac 12, \frac 12 \big)^2 \setminus B_R(0)$ for some sufficiently small $R > 0$. I would like to prove the existence of a solution $\rho = \rho (x_1, x_2)\in C^1(U)$ to the ...
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0answers
65 views

Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$ ...
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1answer
114 views

Does this PDE have a name?

I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$: $ u_{xy} = u_x e^u + u_y e^{-u} $ e.g., Does it have a name? Is it known ...
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149 views

Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
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2answers
306 views

Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
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1answer
117 views

Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs: Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
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1answer
262 views

upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
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44 views

Global Harmonic Oscillator

My question essentially is how to find the appropriate functional space to study uniqueness of solutions to a specific pde. Consider the following pde in three dimensions globally: $ -\tau^2 \...
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2answers
176 views

Is there a true many-body green's function for interacting systems?

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition: ...
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2answers
242 views

A generalization of holomorphic functions

Assume that $U$ is an open set in the complex plane $\mathbb{C}$ and $A$ is a real $2\times 2$ matrix. We define $$\mathcal{S}_{A}=\{f:U\to \mathbb{C}\mid Df.A=A.Df \}$$ where $Df$ is the $2\...
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1answer
339 views

Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
2
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0answers
40 views

Leray-Ohya Hyperbolic System of PDEs

I have a second order PDE system. I calculated symbol of the operator and considered the determinant. This determinant is not a hyperbolic polynomial with respect to any vector in R^4. I came across a ...
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72 views

Imposing boundary conditions and self-similarity on a PDE

This question is an exact duplicate of the question Imposing boundary conditions AND self-similarity on a PDE posted by Stan Corey Carter on math.stackexchange.com. I have a PDE in the ...
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1answer
129 views

Does $u_{t}=g(t)u_{x}^{2}$ blow-up for bounded positive g? What about $u_{t}=u_{xx}+g(t)u_{x}^{2}$?

My original problem is to see if the following pde develops blow-ups in $(-L,L)$ $$u_{t}=u_{xx}+g(t)(u_{x})^{2}$$ for periodic boundary $u_{0}(-L)=u_{0}(L)$, where $0<g(t)<1$; specifically $g(...
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12 views

Setting bound on particular integral when proving properties of Bogovskii operator

I am reading the proof of the properties of Bogovskii operator in the book Introduction to the Mathematical Theory of Compressible Flow. Let $B^\epsilon(y) = \{ x : |x-y| > \epsilon \}$, $f$ and $...
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1answer
73 views

why is paraproduct or paradifferential calculus important in PDE theory?

In the article https://www.baidu.com/link?url=W1BjGmDoZM8QkrV_Qd_26vzNhCJGWyfH79q5cn7q0QQxomVLtH7Fw_mApElkfCZUWiDcYjNhoLhMrGFEXtf4O_&wd=&eqid=a93906890002f93700000003577cbb98, it says that "......
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31 views

The jump set of $SBV$ function for different value of parameter in image denoising problem

The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary, $$ u_\...
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0answers
76 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
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66 views

Elliptic regularity on the hypercube

Assume $$ Lu=f\quad \text{in } [0,1]^d\\ u=0 \quad\text{ on } \partial[0,1]^d $$ for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the ...
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0answers
52 views

Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference or hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution to ...
4
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2answers
148 views

Recover Embedding from Metric

Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known. And suppose that I know ...
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2answers
967 views

Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde: $$\partial_{[\nu}\Gamma^\kappa_{\mu]\lambda}+\Gamma^\kappa_{[\nu|\rho|}\Gamma^\rho_{\mu]\lambda}=\frac{1}{2}R_{\mu\nu\...
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1answer
142 views

Boundary conditions for Klein-Gordon equation

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...
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1answer
440 views

Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...
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Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE. I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of $$\inf_{a \in [-1,1]} \{...
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45 views

Conservation of charge and energy in the Schrödinger equation

In Cazenave's Semilinear Schrödinger Equation, page 56, he describes derivation of conservation of charge and energy of the equation $iu_t+\Delta u+|u|u=0$, ($\alpha=\lambda=1$ and $n=3$, if referring ...
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496 views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}...
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A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation: $$ -\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$ Here $a_i:...
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I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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51 views

Conservation laws for modified Degasperis-Procesi equation

It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, $$E(u)=\frac{1}{2}\int_{0}^{...
5
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1answer
75 views

Decay estimates for wave and Klein-Gordon equation in “generic” curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...
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58 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
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1answer
99 views

$H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
35
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8answers
4k views

Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...