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Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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39 views

Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators? In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
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0answers
46 views

Uniqueness of solution of linear PDE of first order

Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$ be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation $$\partial_t \vec u(x,t)=\sum_{j=1}^...
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1answer
97 views

Existence of unique critical points to second order elliptic PDEs

Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1) $$ \begin{cases} Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
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0answers
15 views

Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
4
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1answer
122 views

Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad. I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
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0answers
26 views

Change of variable for the Stokes equations

I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741). Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...
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0answers
46 views

How to solve this linear Cauchy Problem

within my thesis, I am struggeling with the following PDE: $u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$ $u(T,x,y)=1,$ where $a,b,c,d,e,f$ are polynomials and the ...
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31 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
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1answer
47 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
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0answers
31 views

What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?

Let the generator $L$ of a diffusion process be given in Hörmander form, i.e. $$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$ where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...
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2answers
118 views

Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
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0answers
69 views

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\...
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0answers
116 views

Underdetermined PDE

Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ...
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1answer
112 views

Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation $$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$ where $f$ and $1-c$ ...
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1answer
72 views

System of first order linear coupled PDEs [closed]

I'm in trouble finding the solution of this system of 2 PDEs: \begin{equation} \frac{\partial u_1}{\partial t} + a_1 \frac{\partial u_1}{\partial x} = b (u_1-u_2)\\ \frac{\partial u_2}{\partial t} + ...
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2answers
263 views

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 ...
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1answer
90 views

schauder regularity heat equation

Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$. It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
5
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1answer
121 views

The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
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0answers
40 views

Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ : $$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
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1answer
51 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
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0answers
134 views

Critical spaces and energy estimate in NS equation [closed]

There is a ‘rescaling transformation’ that is particularly significant for the Navier–Stokes equations when they are posed on the whole space, but is also important in the local regularity theory. ...
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3answers
313 views

What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
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1answer
121 views

Is fractional Laplacian invariant under rotation?

If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...
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0answers
74 views

Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case \begin{...
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0answers
31 views

Regularity of Poisson problem with rough coefficients and mixed boundary conditions

Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
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1answer
105 views

Question on expansion into Neumann eigenfunctions

Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \...
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0answers
56 views

Why does a homogeneous first-order linear PDE have $n-1$ functionally independent solutions? [closed]

Why does a homogeneous first-order linear PDE$$\sum_{i=0}^n\xi_i(\mathbf{x})\frac{\partial u}{\partial x_i}=0,$$where $\mathbf{x} = (x_1, x_2, …, x_n)$, have $n-1$ functionally independent solutions ...
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0answers
89 views

What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?

Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...
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1answer
77 views

How do I show continuity of the mixed and weak solution to Zaremba problem?

I am interested in showing continuity/boundedness of the weak solution to the following problem pde: \begin{align*} 0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\ 0 &= \...
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0answers
81 views

Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post): $$\mathcal{A}p=0, \quad p\in C^2(\...
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1answer
329 views

Is this function positive?

Could someone tell me if my argument is correct? Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...
5
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1answer
221 views

Existence of second order potential for PDE

There is a statement in the literature (see the paragraph between equations (18) and (19) in http://aip.scitation.org/doi/10.1063/1.523863), which I would like to generalise, but I don't have a nice ...
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0answers
96 views

Maximum principle on noncompact manifold with boundary

On a complete noncompact manifold with ricci curvature bounded below, we have Yau's generalized maximum principle. What if we have we have noncompact manifold with compact boundary and one complete ...
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3answers
188 views

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
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0answers
37 views

Nonautonomous wave equation of memory type

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation $$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$ This problem can be written ...
3
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2answers
166 views

General formula for integrating factor of an homogeneous differential 1 form

This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
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1answer
91 views

Poisson Equation across a Hypersurface [closed]

Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem $ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
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0answers
198 views

Laplace problem with Robin boundary condition on a wedge

I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let \begin{equation*} \Omega = ...
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0answers
133 views

Uniqueness in a linear elliptic PDE

Let $a \in L^{\infty}(\Omega)$ and suppose $a$ is continuous near the boundary $\partial \Omega$. Assume $w\in H^1(\Omega)$ solves $\Delta w=a w$ in $\Omega$ with $w=\frac{\partial w}{\partial \nu}=...
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1answer
80 views

Decay time to constant function of heat kernel on 2-sphere

Let us consider solving the heat equation on the sphere given a delta function as initial data. $$(\partial_t - \Delta)K(x,y;t) = 0 $$ $$K(x,y;0) = \delta(x,y)$$ One would expect that for large ...
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1answer
124 views

Growth at infinity of a solution to a parabolic PDE

Let us consider the equation: \begin{align*} (\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\ u(0,x) & = u_0 \end{align*} defined on the whole real line (so in one dimension - but ...
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2answers
443 views

Orthogonality to harmonic functions

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define $a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$ $b_n=f\Delta^{...
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1answer
79 views

Examples of the time-dependent linear wave equation

I am looking for examples of the non-autonomous linear wave equation that have some relevant applications. What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial ...
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1answer
50 views

Improved maximum principle estimates (deleting first mode)

Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write $$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$ where $ r=|x|$ and $ \theta = \frac{...
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0answers
47 views

An existence result for solutions of elliptic equations with a mixed boundary problem

Assume that $\Omega$ is a bounded domain such that $\partial\Omega=\Gamma_1\cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint and closed. Let us consider the following elliptic equations. ...
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1answer
541 views

Geometry of curves on the sphere

Let P be a finite set of points on the unit sphere $S^2$ such that for every $p\in P$, there exists a closed curve $\gamma_p \subset S^2$ which has a self intersection at $p$ and passes through $-p$....
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1answer
219 views

Functions orthogonal to harmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose $\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
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0answers
95 views

Continuous right inverse to the Laplacian operator on $C^\infty$

For each $f\in C^\infty(\mathbb R^n)$, there exists $u\in C^\infty(\mathbb R^n)$ such that $\Delta u=f$. This, I guess, has been known well before the more general Malgrange-Ehrenpreis theorem that ...
3
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0answers
69 views

Partial regularity for transmission problem in corner domains

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
3
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1answer
96 views

For an arbitrary $G(x,t)$, does $f_t=2G_xf+Gf_x$, $f(x,0)=0$ have a unique solution for $f$?

I asked the following question at Math Stackexchange a while ago here but did not get a correct answer. Let $f(x,t)$ and $G(x,t)$ be smooth functions from $\mathbb R^2\to\mathbb R$. The ...