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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Discovery of norm in PDE

We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,\alpha}$. Roughly speaking, under appropriate norm, we could capture the ...
89085731's user avatar
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Maximum principle for an elliptic like operator

I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something ...
Math604's user avatar
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Simple existence and uniqueness for second order and linear elliptic PDE

Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$. I am interested on the existence of solution for the following problem: given a continuous ...
L.F. Cavenaghi's user avatar
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Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
Matthias Ludewig's user avatar
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improvement of flatness in the regularity of minimal surfaces

Recently,I am reading Savin's celebrated theorem about improvement of flatness in proving the regularity of minimal surfaces. I have some questions. 1.How to show the boundary of a minimal set ...
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Reference for De Giorgi-Nash-Moser theory

I am interested in Holder regularity for equations of the form $$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic. This was proved in the seminal paper of John Nash ...
Adi's user avatar
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Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions

A function $f$ is Log-Lipschitz if there exists a constant $C >0$ such that \begin{equation} |f(x) - f(y)| \le C|x-y| |\log|x-y|| \end{equation} I am trying to construct two functions with the ...
Rahul Raju Pattar's user avatar
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A problem about closed 2-forms on Minkowski space

The problem is: For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz ...
jacktang1996's user avatar
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References on thin film equation: derivation and properties

Where can I find a derivation of the thin film equation $$u_t = - \mathrm{div} (u^m\nabla\Delta u)$$ from a physical model? a good introduction to its properties (e.g. conserved quantities and ...
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Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following

Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^...
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How to evolve a star-shaped mean convex set to a strictly mean convex set?

I have trouble in going through a proof in a paper for quite a while. To simplify notation and make it readable to a larger audience, let me just present the simplest case: Let $\Omega$ be a bounded ...
student's user avatar
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Uniqueness of viscosity solutions of Hamilton-Jacobi equation

Consider the following Hamilton-Jacobi (HJ) equation: $$u_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n ...
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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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Symplectic formulation of compressible Euler equation

It has been widely known that the compressible Euler equation can be cast into the Hamiltonian form. For example, in the book "Dubrovin B A, Fomenko A T, Novikov S P. Modern geometry—methods and ...
sam's user avatar
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Limits for eigenvalues for the Dirichlet Laplacian

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem $$ \begin{cases} -\Delta u=\lambda u & \mbox{in }\Omega\\ u=0 & \mbox{on }\partial\...
de Araujo's user avatar
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Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
user57055's user avatar
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To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function such ...
Inquisitive's user avatar
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Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
TheBook's user avatar
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on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
Delio Mugnolo's user avatar
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Norm of differential operator between Sobolev spaces

It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms) $W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
Sylvester-H's user avatar
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Is this kernel space of finite dimension ?

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
user23078's user avatar
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The extension of smooth function

If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
Adterram's user avatar
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Finding an $H^1$ function given its values on $\partial\Omega$

Background I've met this problem when I was trying to convert a elliptic PDE problem into the corresponding variational problem in order to apply finite element method. The PDE is an elliptic PDE ...
Roun's user avatar
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A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e

I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is not differentiable a.e. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,...
Dorian's user avatar
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Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.

A friend of mine in the department needs to know if the following PDE has been extensively studied $$ u_t = (u^2)_{xx}$$ Or more generally, replacing the square by any function of $u$. One would like ...
John Jiang's user avatar
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$L^{\infty}$ estimate for heat equation with $L^2$ initial data

Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{...
Bogdan's user avatar
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Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?

I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...
Talmsmen's user avatar
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$W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations

Let $B$ denote the $n$-dimensional unit ball. Assume $u\in\bigcap_{1\le p<2} W_0^{1,p}(B)$ satisfies $$ \int a_{ij}\partial_j u \partial_iv =0,$$ for any $v\in C_c^{\infty}(B)$, where we assume $a_{...
Tian LAN's user avatar
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A detail in one step in a theorem from a paper of Brezis and Merle

In paper [1] Brezis and Merle prove theorem 3 by using the following fact. Let $w_n=u_n-v_n$, $\Delta w_n=0$ on $\Omega$ (a bounded domain in $\mathbb{R^2}$) and $w_n^+$ is bounded in $L^{\infty}_\...
User1132's user avatar
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134 views

Strichartz estimates and Schrödinger equation with derivative

Let $$ iu_t + \Delta u = \Phi \cdot \nabla u$$ with initial data $u_0 \in H^s(\mathbb{R}^d)$ and $\Phi\colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ sufficiently regular. Suppose I want to use ...
Jakob Möller's user avatar
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2 answers
221 views

Change of variables for obtaining a unitary group

Consider the following NLS: $$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$ where $F(u):=(u + \bar{u} + |u|^2)u.$ In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
Mr. Proof's user avatar
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1 answer
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Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
Zac's user avatar
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1 answer
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Solvability of general linear PDE with constant coefficients

Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$. Is it true that the equation $$Du=f$$ is solvable in any ...
asv's user avatar
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1 answer
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Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$. What i want to ask is, if $S_C$ is nonempty for some $...
W.J.'s user avatar
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1 answer
403 views

Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
Mr. Proof's user avatar
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1 answer
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About the proof of higher regularity boundary Harnack inequality

I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1. In the paper they used the Hopf lemma to show that $u_\nu>c&...
user734979's user avatar
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1 answer
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Linear transport equation with Lipschitz conditions

Given the equation here, I would like to ask the following relaxed question: Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = ...
Pritam Bemis's user avatar
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1 answer
334 views

Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?

I should preface this question by saying that I strongly suspect the answer is negative, but I couldn't find the counterexample myself. Say we are working on the unit disc $D \subset \mathbf{R}^n$, ...
Leo Moos's user avatar
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On the equation $[U, V] - V_x = C(x)$

While considering the zero curvature equation $U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ ...
Talmsmen's user avatar
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1 answer
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A possible error in Villani's monograph "Hypocoercivity"

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\...
Fei Cao's user avatar
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1 answer
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wave equation with vanishing trace at infinity

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem \begin{equation}\label{pf0} \begin{aligned} \begin{cases} \Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
Ali's user avatar
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Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
Giovanni Febbraro's user avatar
3 votes
1 answer
129 views

Smoothing-Strichartz estimates for the heat-Schrodinger evolution

Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the ...
Capublanca's user avatar
3 votes
1 answer
695 views

Continuous extension of functions

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and $f \in W^{1,p} (\partial \Omega)$. Can $f$ be extended to a function $u \in W^{1,p}(\Omega)$ such that $u|_{\partial \Omega}=f$ and $$\lVert u\...
MathLearner's user avatar
3 votes
1 answer
581 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
3 votes
1 answer
204 views

Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
Sascha's user avatar
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1 answer
245 views

Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
R Mary's user avatar
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1 answer
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Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
Junhyeong Kim's user avatar
3 votes
1 answer
316 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 ...
Onil90's user avatar
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1 answer
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Gagliardo-Nirenberg inequality for bounded domain

For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that $$ \|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}. $$ This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
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