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Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): 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 votes
1 answer
142 views

The Beltrami equation and Neumann series

Let $\mu: \mathbb{C} \to D(0,1).$ A quasiconformal map is a $W^{1}_{2,loc}-$solution to the Beltrami equation $\bar{\partial}f = \mu \partial f$. In this paper, the authors remark that one can ...
4 votes
0 answers
136 views

Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
5 votes
0 answers
140 views

Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
2 votes
0 answers
728 views

Proof of the link between the Fokker–Planck equation and SDE?

I know the link between the Fokker–Planck equation and SDE given by the Feynman-Kac theorem is as follow: $$d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}$$ $$\frac{\partial}{...
5 votes
1 answer
359 views

Recovering the nonlinear Schrödinger equation from its Lax pair

My question is less concerned with the physical aspects of the nonlinear Schrödinger equation and more with the mathematical mechanics of using a Lax pair. I am considering how to recover the ...
7 votes
1 answer
281 views

Existence of harmonic maps onto the $n$-sphere

Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g_{st})$ be the standard round $n$-sphere. Whether there exists a non-...
6 votes
1 answer
182 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
6 votes
0 answers
151 views

Gap between consecutive Dirichlet eigenvalues

Suppose $\Omega \subset \mathbb R^2$ is a domain with a Lipschitz boundary and let $\{\lambda_k\}_{k=0}^n$ be the eigenvalues for the Laplacian operator on $\Omega$, that is to say $$ -\Delta \phi_k = ...
5 votes
1 answer
445 views

Schwartz regularity for the density of a stochastic process

Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables $$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$ It ...
2 votes
1 answer
130 views

Canonical forms on higher degree Jet bundles similar to the Liouville form

On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^*(X)$, to the Hamiltonian mechanics is well known; $T^*(X)$ is a degree $1$-Jet bundle. ...
1 vote
0 answers
31 views

What does the placeholder in the expression means in the paper "Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem"?

I am a researcher whose research interest is in applying numerical algorithm to solve inverse problems. In reading the paper Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem, I ...
0 votes
1 answer
98 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
3 votes
1 answer
218 views

Are there applications for the PDE $ - \operatorname{grad} ( \operatorname{div} \vec u ) = \vec f$?

As in the title: given a vector field $\vec f$, are there any interesting applications (in physics, biology, or economy, or ...) of the partial differential equation $ - \operatorname{grad} ( \...
1 vote
0 answers
53 views

Maximum principle for a fractional Laplacian operator

Let $\Omega=\{x\in \mathbb R^N: R_1<|x|<R_2\}$ be a bounded in $\mathbb R^N$ and $N>2s$ with $s\in (0, 1)$. If $u$ is a smooth solution to the problem $(-\Delta)^su=0$ in $\Omega$ with $u=f$ ...
2 votes
0 answers
113 views

Divergence-free constraint for a boundary integral equation

Consider the system $$ \begin{cases} \operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\ c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
1 vote
0 answers
33 views

Pointwise estimate of solutions to the parabolic equation with a monotonic drift

I wonder for a parabolic equation $$u_t+(a(t,x)u)_x= u_{xx},$$ if we know that $a(t,x)$ is monotonic decreasing in $x$ with $a(t,-\infty)=C_L, a(t,+\infty)=C_R$, $C_L>C_R\geq 0$, are there results ...
4 votes
1 answer
377 views

Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
4 votes
0 answers
146 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
6 votes
0 answers
114 views

Determine the location of the boundary where the heat changes fastest

I am motivated by the following question: Given a uniform heat source in a convex domain, and suppose that the outside temperature is equal to $0$, can we determine where the long-time temperature ...
16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
2 votes
1 answer
160 views

Sobolev imbedding result; proof

Here I would like to prove a result that I assume is known but I am having difficulty proving. I will give the set up. This problem is really coming from functions with are 'doubly radial' or '...
2 votes
0 answers
658 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
1 vote
0 answers
103 views

A PDE involving a diffeomorphism of $\mathbb{S}^1$

This question is a special case of this one. Let $s(\theta)>0, b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$. Do there exist a diffeomorphism $\phi:\mathbb{S}^1 \to \...
2 votes
1 answer
289 views

Reference request for semilinear PDEs in dimension 2

I am interested in the study of the (semi-linear, I suppose) equation $$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\ u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$ ...
2 votes
0 answers
42 views

convergence in capacity implies $\gamma$-convegence

Let $A_n$ a sequence of quasi-open sets and $A$ a quasi- open set contained in a ball $B$ in $\mathbb{R}^N$. Let $u_A$ the unique solution in $H_0^1(A)$ of the equation $-\Delta u=1$. It means that $$ ...
2 votes
0 answers
84 views

Decay rate of transition density of a SDE system

Consider the following SDE system $$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$ Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...
6 votes
2 answers
340 views

Vacuum region with positive measure for the Schrödinger equation

Let us consider the free Schrödinger equation $(i\partial_t+\Delta_x)\psi=0$ in $\mathbb{R}_t\times\mathbb{R}_x^d$. I'm trying to understand the structure of the vacuum region $$\Omega(\psi):=\{(t,x)\...
0 votes
0 answers
244 views

Holder continuity relative to Rellic-Kondrachov compactness via the nonlinear Aronsson operator

Connected to the question, Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness? An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ ...
0 votes
2 answers
291 views

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: ...
1 vote
0 answers
105 views

Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?

Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
3 votes
0 answers
189 views

Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator. Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
5 votes
0 answers
156 views

Homotopy, contraction mapping and the inverse function theorem on Banach spaces

We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
2 votes
0 answers
144 views

Does this geometric PDE have a solution?

Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes. Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
0 votes
1 answer
142 views

Showing that a Gaussian achieves equality in a logarithmic Sobolev inequality

Consider the following Logarithmic Sobolev inequality on page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14): for $f\in H^1(\mathbb R^n),$ and $a>0$ any positive number, $$ \frac{a^2}{...
1 vote
1 answer
275 views

Regularity of solutions for a non linear elliptic equation

Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$ $(-\Delta)^2 v_k=e^{v_k}$ $v_k(x)\leq v_k(0)=0$ $\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad R&...
0 votes
0 answers
982 views

Weak $H^1$ convergence implies strong $L^p_{\text{loc}}$ convergence

On page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14), there is a theorem stating that if $f_j \to f$ weakly in $H^1(\mathbb R^n)$ then $\mathbf 1_A f_j \to \mathbf 1_A f$ strongly in $...
2 votes
1 answer
257 views

Existence of divergence-free unit vector field in conformally rescaled euclidean metric

Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
-1 votes
1 answer
206 views

A-priori bound on parabolic PDE that doesn't depend on end time

I have a PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$ where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound (...
0 votes
0 answers
93 views

Regularity of semilinear parabolic PDE in the whole space

I need regularities in Holder space of the following parabolic PDE: $$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
5 votes
1 answer
568 views

Convergence of discrete Laplacian to continuous one

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues ...
4 votes
1 answer
387 views

Asymptotic formula for fractional Laplacian

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan ...
1 vote
0 answers
50 views

Strong convergence from elliptic equation

Consider the following equation $$ \Delta A_n = -2\Im(\bar{u}_n(\nabla -iA_n)u_n) =: -J(u_n,A_n)$$ and suppose that $u_n \in L^{\infty}(I,H^1(\Omega))$ for some bounded $I\subset \mathbb{R}$, $\Omega\...
1 vote
0 answers
323 views

Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?

I have been reflecting on this question, and want to share my thinking thus far. I'd be grateful for the community's inputs. We refer to Morrey's inequality, Theorem 4 on pp 266 of Evan's book on PDE, ...
4 votes
1 answer
266 views

Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$

Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
5 votes
1 answer
721 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}{\left|\Omega\right|} \int_{\Omega} u\|}_{L^2}\leq C {\|\nabla u ...
6 votes
0 answers
113 views

A continuity argument for a dispersive $gKdV$ estimate

I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The ...
0 votes
1 answer
147 views

The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]

Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
5 votes
1 answer
301 views

$L^p$-estimates for elliptic pseudodifferential operators

Assume we have an pseudodifferential operator $P:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S}(\mathbb{R}^n), Pf(x) = (2\pi)^{-n/2}\int\mathrm{d}\xi\; p(x,\xi)\,\hat{f}(\xi)e^{i\xi x}$ acting on ...
27 votes
2 answers
8k views

Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
4 votes
1 answer
1k views

Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...

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