# 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.

2,558
questions

**3**

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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 ...

**0**

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**0**answers

27 views

### 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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140 views

### 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 - \...

**3**

votes

**1**answer

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 (...

**2**

votes

**0**answers

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}
...

**2**

votes

**0**answers

52 views

### 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 ...

**0**

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**0**answers

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 = ...

**1**

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**0**answers

102 views

### Green function of a fractional operator

What is the Green function of the following operator with homogeneous Dirichlet boundary condition?
$$(-\Delta)^s - k \frac{u}{|x|^{2s}} \quad (k\ge 0) $$

**0**

votes

**0**answers

94 views

### 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 ...

**2**

votes

**1**answer

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^\...

**2**

votes

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123 views

### 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 ...

**4**

votes

**0**answers

129 views

### 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 $\Omega=\{a<|x|<b\}.$

**0**

votes

**1**answer

96 views

### 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 ...

**0**

votes

**0**answers

79 views

### 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 ...

**1**

vote

**0**answers

158 views

### Replacing the initial conditions for a PDE

The problem
The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$
$$\partial_t \psi = i \partial_x^2 \psi$$
$$\psi(x,t=0) = \psi_0(x)$$
$$\...

**0**

votes

**1**answer

93 views

### 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'...

**2**

votes

**2**answers

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 ...

**10**

votes

**0**answers

109 views

### 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}^...

**3**

votes

**1**answer

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 $C=C(\kappa,r,m)$(...

**0**

votes

**1**answer

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)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...

**0**

votes

**0**answers

32 views

### 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^\...

**3**

votes

**0**answers

151 views

### 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 ...

**5**

votes

**1**answer

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{...

**1**

vote

**1**answer

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
$$\Delta u=f,$$
...

**2**

votes

**0**answers

33 views

### 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|_{\...

**2**

votes

**1**answer

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|_{\...

**7**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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=-\...

**1**

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**0**answers

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\...

**1**

vote

**2**answers

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 ...

**0**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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?

**0**

votes

**0**answers

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 \...

**1**

vote

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

73 views

### 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|...

**4**

votes

**0**answers

83 views

### 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 ...

**0**

votes

**0**answers

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?

**2**

votes

**1**answer

79 views

### 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)...

**5**

votes

**1**answer

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.
...

**2**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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?

**2**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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 } \...

**3**

votes

**1**answer

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)?

**2**

votes

**1**answer

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**

votes

**0**answers

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 \...

**1**

vote

**0**answers

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 ...

**2**

votes

**0**answers

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 (|\...

**1**

vote

**0**answers

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{\...