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

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16
votes
0answers
433 views

Why is there a connection between enumerative geometry and nonlinear waves?

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it. Recently I encountered in a class the fact that there is a generating function of ...
11
votes
0answers
354 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
9
votes
0answers
117 views

Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates ...
8
votes
0answers
363 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
7
votes
0answers
148 views

Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$. C.D. Sogge proved that we have the ...
7
votes
0answers
312 views

Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble : Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function ...
7
votes
0answers
247 views

Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
6
votes
0answers
113 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
6
votes
0answers
325 views

Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
6
votes
0answers
264 views

Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ...
6
votes
0answers
272 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...
6
votes
0answers
369 views

How to estimate isoperimetric constant?

Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties, There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$ for ...
6
votes
0answers
238 views

Dirichlet-to-Neumann map on $C^{k,1}$ domains

I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in ...
6
votes
0answers
767 views

Harmonic maps into compact Lie groups

Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck, Schoen, Richard; Uhlenbeck, ...
6
votes
0answers
356 views

A non-elliptic PDE

I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X : $\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial ...
5
votes
0answers
91 views

Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that $||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...
5
votes
0answers
80 views

$L^p$ regularity for wave equations with coercive boundary conditions

Suppose we have the wave type equation $$\partial^2_tu - L u = 0$$ on a compact manifold with boundary, where $L$ is a second order strongly elliptic operator with coercive boundary conditions (not ...
5
votes
0answers
106 views

A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$. Let now ...
5
votes
0answers
269 views

Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
5
votes
0answers
141 views

Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: ...
5
votes
0answers
93 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
5
votes
0answers
327 views

“Euler system” in Christodoulou's The Action Principle and PDEs

In The Action Principle and PDEs Christodoulou spends some time describing what he calls the Euler system associated to a system of variational PDEs (sections 2.5-7, 6.2). Briefly, given a bundle ...
5
votes
0answers
443 views

Any similar inequality in literature?

I got the following inequality: $B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
4
votes
0answers
125 views

Is the universal constant in Caccioppoli's inequality one?

If you go through the classical proof of the fact that if $\Delta u= 0$ on $B_R$, then for every $R^\prime<R$ there holds $$ \int_{B_{R^\prime}}|\nabla u|^2 \leq ...
4
votes
0answers
80 views

Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...
4
votes
0answers
101 views

Carleman estimates on monotonicity formulas

I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from $$\int_{B_r} ...
4
votes
0answers
224 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
4
votes
0answers
146 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
4
votes
0answers
200 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
4
votes
0answers
107 views

Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question: Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$: $$ \frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t), $$ with smooth initial data ...
4
votes
0answers
312 views

Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction ...
4
votes
0answers
341 views

Existence of solutions to elliptic PDE in undbounded domain

Specifically, I have $Lu=f$, where $L$ is a linear elliptic pseudodifferential operator, on an unbounded domain of the form $\Omega\times [0,\infty)$, $\Omega$ has Lipschitz boundary, $u$ is 0 on all ...
4
votes
0answers
764 views

A “Cauchy integral formula” for the Poisson kernel?

The inspiration for this question is in a certain breakdown of the analogy between holomorphic and harmonic functions. First recall the Cauchy integral formula: Let $U$ be an open subset of ...
4
votes
0answers
129 views

Uniqueness for solution of a d-dbar system related to Davey-Stewartson Solitons

This question concerns a system of equations that arise in the study of one-soliton solutions to the Davey-Stewartson equation. In what follows, $f(z)$ denotes a function which depends smoothly (but ...
4
votes
0answers
104 views

Regularity of reflection coefficients (or more generally the scattering transform)

Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$. We define the ...
3
votes
0answers
36 views

Is Besove spaces $B^{s}_{p,q}$ invariant under Fourier transform?

(This may be very easy question for MO; as I am just trying to understand Besov spaces) Let $\phi \in C^{\infty}(\mathbb R^{n})$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: ...
3
votes
0answers
75 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
3
votes
0answers
112 views

$\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
3
votes
0answers
62 views

Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
3
votes
0answers
141 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
3
votes
0answers
85 views

Linear heat equation with initial condition of generalized function

I am consider a very simple heat equation over the interval $[0, 1]$ with a Neumann BC and a very bad initial condition, written as: $\partial_tu(t, x) = \partial^2_xu(t, x) + a(t, x)u(t, x)$, for ...
3
votes
0answers
171 views

Reference on a Monge-Ampère-like equation

We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$: $$ \det(Hess(w))=1~, $$ ...
3
votes
0answers
95 views

Subordination identity and heat operator

I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...
3
votes
0answers
86 views

$L^2$ bounds for the gradient of subsolutions to parabolic equation

Suppose we have the differential inequality $$ |\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|) $$ in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...
3
votes
0answers
193 views

Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
3
votes
0answers
204 views

is this a known method for solving PDEs

I recently posed a system of PDEs to solve on MSE at http://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions. However, in this post, I'd like to show here ...
3
votes
0answers
138 views

W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...
3
votes
0answers
98 views

Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation $- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin ...
3
votes
0answers
134 views

How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded

With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider $$u' + Au = f$$ in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
3
votes
0answers
163 views

Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...