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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100 views

Singular integral operators and PDEs

What is the relation between the notion of singular integral operators and partial differential equations? I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
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Example/Reference needed for Laplace equation coupled with another equation

I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two ...
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Suitable ansatz to the system of PDEs

I have the following three PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \...
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1answer
221 views

Solution to a nonlinear PDE

A stochastic control problem has led me to the following PDE: State space: $t \in [0,T]$ and $x \in [-1,1]$. $$4 \frac{\partial f}{\partial t} \frac{\partial^2 f}{\partial x^2} = 1 \quad \forall (...
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56 views

discrete parabolic Harnack inequality

I am currently looking for a discrete version of the parabolic Harnack inequality in the following "$L^1$ to $L^\infty$" form: If $u(t,x)\geq 0$ is a (say, smooth) subsolution of \begin{equation} ...
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Differences among various index theories in critical point theory

Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones? the ...
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225 views

Local “boundary comparison principle” for harmonic functions

Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
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102 views

Green function of a fractional operator

What is the Green function of the following operator with homogeneous Dirichlet boundary condition? $$(-\Delta)^s - k \frac{u}{|x|^{2s}} \quad (k\ge 0) $$
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
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1answer
228 views

Density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right), $$ $$ H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}. $$ Q: Is $C^\...
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Variational formulation for elliptic interface problem

Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
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Sobolev embedding in an annulus

Is it possible to determine the best constant $S(\Omega, p)$ of the embedding $H^1_{0, r}(\Omega)$ to $L^{p+1}(\Omega)$ where $p>1$ and $\Omega=\{a<|x|<b\}.$
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1answer
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Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
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Strong continuity (weak to strong) of $\langle Au,v\rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the Sobolev space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v\rangle=\int u^3 v dx$$ is strongly (weak ...
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158 views

Replacing the initial conditions for a PDE

The problem The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$ $$\partial_t \psi = i \partial_x^2 \psi$$ $$\psi(x,t=0) = \psi_0(x)$$ $$\...
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1answer
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Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...
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2answers
123 views

2d incompressible Euler equations under periodic boundary conditions

It is well known that the 2d incompressible Navier-Stokes equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions. I tried searching for a ...
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A geometrical problem in terms of a convex function

I wish to know whether the following problem has ever been investigated. Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
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1answer
215 views

Local Sobolev embedding on complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball. Q Can we find a constant $C=C(\kappa,r,m)$(...
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1answer
221 views

Stone–von Neumann theorem?

The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR) $$ U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t $$ ...
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Reference request: Numerical methods for Hamilton-Jacobi-Bellman equations with state constraints

I have two questions on numerical methods for solving Hamilton-Jacobi-Bellman (HJB) equations with state constraints. Consider an optimal control problem given by $$ v(x) = \max_{\{u(t)\}_t} \int_o^\...
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Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
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1answer
270 views

Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average{...
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1answer
123 views

Poisson equation on noncompact manifold

Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space. For the equation $$\Delta u=f,$$ ...
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Quasilinear elliptic problem on fractal domain

Consider the following quasilinear elliptic equation $$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$ on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\...
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1answer
115 views

Quasilinear elliptic problem: Ellipticity-type conditions

Consider the following quasilinear elliptic equation $$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$ on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\...
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1answer
173 views

Almost orthonormal projection and orthonormal projection in Hilbert space

Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e. $$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$ and $\alpha$ is ...
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1answer
371 views

A question of the Schrodinger Semigroup --By B. Simon

The question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3). On the Theorem C.1.2(subsolution estimate) of the paper, it says that: If $Hu=0$, where $H=-\...
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116 views

On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define $$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$ where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
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2answers
134 views

The difference between the nonlocal and local conditions problems

In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed. In this paper: Existence and uniqueness ...
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1answer
54 views

Finding Free surface elevation in semi-infinite channel

A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite ...
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1answer
98 views

Analytic approach to geodesic connectedness in Semi-Riemannian manifolds

Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
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37 views

Anisotropic elliptic problems

Where can I find a reference on the following anisotropic problem? $$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$ where $(x,y) \in \...
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1answer
81 views

Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$. For a model case, consider a ball split in a smaller ball and an anulus. Consider the following elliptic ...
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43 views

Local Cancellation in Real Hardy Space

I want to show the following asymtotic estimate in Hardy space over $\mathbb{R}^n$: Let $a\in \mathbb{R}^n$. I want to show the function $$ f(x)=\mathbb{1}_{B(0,1)}-\mathbb{1}_{B(a,1)} $$ is asymtotic ...
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Biharmonic equation

Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation \begin{equation}\label{\star} \Delta^2u+V(x)u=g(x, u)+K(x)u, \end{equation} where $|g(x,s)|\leq \varepsilon|...
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Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”

I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
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44 views

Existence and uniqueness for semilinear parabolic problem using fixed point approach

Where can I find a proof of existence and uniqueness of solutions for a semilinear parabolic problem $$u_t -\Delta u +f(t,x,u,\nabla u) =0$$ which is based on a fixed point approach?
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1answer
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Well-posedness of wave equations with time-dependent coefficient

Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)...
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1answer
194 views

Obstruction to Navier-Stokes blowup with cylindrical symmetry

Is there a known obstruction to cylindrically symmetric solutions (with swirl) of incompressible 3D Navier-Stokes blowing up in finite time ? EDIT: in the whole space $\mathbb R^3$, I forgot to say. ...
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44 views

Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain

Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
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51 views

Elliptic Dirichlet problems with measure boundary data

Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
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1answer
99 views

Boundary condition for elliptic problems and domain decomposition

This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains Consider an open domain $U$ split in two non-overlapping ...
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1answer
207 views

Elliptic problem on a domain split in two subdomains

Consider the following elliptic problem in a split domain: $$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\ -\Delta u =f_2 & \text{ in } U_2\\ u=g & \text{ on } \...
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1answer
274 views

Heuristics for boundary Harnack inequality

What is the heuristic idea of the proof of the boundary Harnack inequality presented in the appendix of Caffarelli's 1998 lectures on the obstacle problem (page 38 here)?
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1answer
163 views

Bounded solution for parabolic equation

Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\...
3
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0answers
64 views

Green's function of time-dependent Stokes equation

It is well known that the Green's function of a standard parabolic equation in a bounded domain, say \begin{align} \partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in \...
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124 views

Reference request: Proof of Lemma 3.3 in Linear and Quasi-linear Equations of Parabolic Type

The following lemma in Linear and Quasi-linear Equations of Parabolic Type by Nina Uraltseva, Olga Ladyzhenskaya, and Vsevolod A. Solonnikov. Lemma 3.3 Let $\Omega\subset\mathbb{R}^n$ satisfy a cone ...
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182 views

The use of dissipation in parabolic equations

I'm considering an equation in Sobolev spaces and stuck at a dissipation term. After constructing my desired Sobolev norm $W^{s,q}$, on the left hand side of the equation, I have $$\Vert \nabla (|\...
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60 views

Singular integral of the composition of the Hilbert transform and fractional Laplacian

Given $0<s<1$, we can define the Fractional Laplacian by $$\Lambda^{-s}f(x):=(-\Delta)^{-s/2}(x)=\int_{-\infty}^{+\infty}|x-y|^{-1+s}f(y)dy$$ or by means of Fourier transform as $$\widehat{\...