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

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32 views

Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem $$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...
3
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0answers
26 views

On a continuous extension of a linear 2nd order PDE

Consider an elliptic (hyperbolic) equation $A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$ in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least ...
2
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0answers
38 views

Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse. Does there ...
4
votes
1answer
401 views

Beginners Guide to Cartan for Beginners [on hold]

I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance. Question: I am seeking ...
3
votes
1answer
66 views

Application of Egorov's Theorem for Pseudodifferential Operators

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
2
votes
1answer
84 views

System of linear first order PDE with constant coefficients

recently in my researches I've come across the following operator $$L\left(\begin{array}{c} a_1\\ \vdots\\ a_n \end{array}\right)=M_1\left(\begin{array}{c} ...
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0answers
32 views

Existence to an advection equation with constraints

Let $f: \mathbb R^n \to \mathbb R^n$ be a sufficiently smooth vector field and let $h: \mathbb R^n \to \mathbb R$ a scalar field. We consider the following advection equation ...
2
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1answer
111 views

Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation $$ \frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T], $$ subjected to ...
4
votes
1answer
181 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 ...
2
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1answer
88 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
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2answers
128 views

Frobenius condition

Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...
0
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0answers
57 views

How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
1
vote
0answers
73 views

Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...
4
votes
1answer
152 views

Extension of solutions of PDEs with constant coefficients

Let $\mathcal L$ be a differential operator with constant coefficients and $\mathcal{L} f=0$ for some $f\in C^{\infty}(\mathbb{R}^n).$ Under what conditions on $\mathcal {L}$ the function $f$ extends ...
2
votes
2answers
80 views

Estimates on a heat process with fixed boundary data and zero initial conditions

Consider the following heat process: For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that $$ \partial_t ...
0
votes
1answer
92 views

Prove a function, defined by integration of a harmonic function, is log-convex [closed]

Let $u$ be a harmonic function and we define $$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$ The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function ...
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0answers
36 views

Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
1
vote
1answer
42 views

Point moving inside smooth domain?

Let $U \subset \mathbb{R}^2$ be a domain im $\mathbb{R}^3$ with smooth boundary. Let a point move inside $\mathbb{R}^3$ along the smooth curve $x(t)$. We denote by $\mbox{dist}(x(t), \partial U)$ the ...
3
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0answers
87 views

Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters? Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...
0
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1answer
88 views

Complex transport equation

Consider an n dimensional Riemannian manifold with boundary. Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...
9
votes
1answer
199 views

applications of C$^*$-algebras in the field of PDEs

I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
6
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1answer
98 views

Is there a maximum principle for stress in continuum mechanics?

I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...
2
votes
1answer
47 views

Dichotomy for global existence or blow up for solutions of evolution problems

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ ...
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0answers
29 views

ABP estimates for semiconvex functions

Referring to the classical ABP Estimate (Gilbarg-Trudinger Lemma 9.2) I am looking for if such an estimate can be generalized to semiconvex functions. In an article of Trudinger (Comparison Principles ...
1
vote
0answers
31 views

Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)? I am thinking of eg. ...
1
vote
0answers
21 views

Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?

Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...
3
votes
1answer
98 views

Surfaces with specific types of second fundamental form

Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...
5
votes
2answers
135 views

Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
4
votes
1answer
106 views

PDEs on torus $\mathbb T$

(Hope this question is o.k. for MO) I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...
5
votes
2answers
217 views

Poincare-like inequality on compact Riemannian manifolds

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of $$ ...
0
votes
1answer
117 views

Equivalence of two definitions of Sobolev spaces

Good morning, I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space $$ D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...
4
votes
1answer
108 views

Minimal surfaces + Semi-Geodesic Coordinates

Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
3
votes
2answers
106 views

Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...
3
votes
2answers
190 views

Isothermal-related functions in higher dimensions

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help. Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
0
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2answers
171 views

Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
0
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0answers
113 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
3
votes
0answers
87 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in ...
1
vote
0answers
42 views

decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
0
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0answers
60 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
-1
votes
0answers
45 views

A fredholm index associated with two vector fields generating a 2 dimensional foliation

Let $M$ be a compact manifold and $X,Y$ be two independent vector fields on $M$ with $[X,Y]=0$. Let $\mathcal{F}$ be the 2 dimensional foliation associated with the distribution ...
1
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0answers
59 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
4
votes
1answer
96 views

$L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} ...
2
votes
0answers
88 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi ...
2
votes
1answer
241 views

Sobolev Space, “characteristic function” for the weak derivative

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product $$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$ for $u\in H_0^1$ and ...
1
vote
1answer
38 views

Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability: One factor would be the condition number of the approximation operator. The other factor ...
0
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1answer
118 views

Green's function and eigenvalues with multiplicity

Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
1
vote
1answer
175 views

Smooth curves in a Frechet space

Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...
2
votes
1answer
107 views

A Liouville theorem involving an advection term

I am curious whether there are any Liouville theorems for the following pde: $$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$ where $a(x)$ is a smooth bounded vector field ...
9
votes
3answers
201 views

Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement: Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs \begin{align} P_i(\partial / \partial x_1, \dots, ...
4
votes
1answer
125 views

Dependence of solutions on parameters in partial differential equations

In the standard homogenization problem $$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$ the homogenized matrix $A_0$ is given in terms of ...