Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
406 questions
0 votes
0 answers
50 views
Problem about the variational formula for the principal eigenvalue of non self-adjoint 2nd linear elliptical operator
Let $P$ be a second order elliptic operator defined in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n, n \geq 2$. $\lambda_0$ is the principal eigenvalue of the Dirichlet eigenvalue problem $$ ...
- 1,011
1 vote
0 answers
90 views
Applying the Krein-Rutman theorem to analyze the eigenvalue and its eigenfunction of 2nd differential elliptic operator on closed manifold
I want to ask a problem about applying the Krein-Rutman theorem to analyze the eigenvalue and its eigenfunction of 2nd differential elliptic operator on closed manifold. Since for general 2nd linear ...
- 1,011
4 votes
1 answer
283 views
Is there a continuation of the same type to "Partial Differential Equations in the 20th Century"?
From the post Historical developement of analysis and partial differential equations (especially in the 20th century) I found "Partial Differential Equations in the 20th Century" and I do ...
- 189
2 votes
0 answers
154 views
Regularity of eigenfunctions of elliptic operator with piecewise-constant coefficients
Consider the generalized eigenvalue problem: $$ [- \nabla \cdot (D(\mathbf{x}) \nabla) + \Sigma_a(\mathbf{x})] \phi(\mathbf x) = \lambda \Sigma_f(\mathbf x) \phi(\mathbf x)$$ Some specifications: The ...
2 votes
1 answer
152 views
A cosine family that exponentially decays
Let $X$ be a Banach space. A strongly continuous family $(C(t))_{t\in\mathbb{R}}$ of bounded operators on $X$ is called a cosine family if $C(0) = I$, $C(t+s) + C(t-s) = 2C(t)C(s)$ for all $t,s\in\...
- 21
2 votes
1 answer
273 views
Solving the curl curl problem in $C^\infty_c$
Let $\Omega$ be smooth and bounded and simply connected, dimension 3. Let us call $D=(C^\infty_c)^3$ and $D_v=$ all divergence free vectors in $D$. Now consider the problem $$\DeclareMathOperator{\...
- 181
1 vote
0 answers
174 views
Diffusion approximation to the Schrödinger equation with low frequency initial data
I'm trying to figure out if the solutions to the Schrödinger equation and the diffusion equation are close for low-frequency/smooth initial data. I.e., I want to bound $$ \|u(t,x) - v(t,x)\|_{L^\infty}...
- 103
1 vote
0 answers
82 views
A problem about the contact form and the Monge-Ampere system on Riemannian manifold
The definition of contact manifold is: Definition A contact manifold $(M, I)$ is a smooth manifold $M$ of dimension $2 n+1\left(n \in \mathbf{Z}^{+}\right)$, with a distinguished line sub-bundle $I \...
- 1,011
0 votes
0 answers
48 views
Some identitites for general linear elliptic equations
I am looking for interesting (or a list) identities for solutions of $-\text{div} A(x)\nabla u=0$. Two examples I have are the Pohozaev (or Rellich) identity and Bochner identity. I am curious to find ...
- 537
-5 votes
1 answer
240 views
How to compute the derivative of an integral with respect to a function? [closed]
Let $\bar{x}(t) = \int_c^d x(t, \alpha) p(\alpha) d\alpha$, with $p$ a given function and $c,d$ constants. I want to compute the derivative of $\bar{x}$ with respect to $x$ Edit: I noticed that my ...
- 65
0 votes
0 answers
67 views
Automatic balancing of multiple loss terms in polynomial PDE optimization
I'm solving an inverse PDE problem by fitting a signed distance function (SDF) to a 2D shape using polynomial approximation (Chebyshev basis). The loss consists of: a boundary loss: $ u = 0 $ on the ...
- 21
4 votes
1 answer
199 views
How to find an image of the Lie bracket?
I am studying a certain equivalence of function locally around $x=0$. For a fixed map (vector field) $f\in C^\infty (\mathbb{R}^n, \mathbb{R}^n)$, I am computing the tangent space to its class of ...
- 41
0 votes
1 answer
153 views
Embedding of Sobolev spaces with the weight $|x|^\tau$
Let $\Omega \subset \mathbb{R}^N$ a bounded domain. For $\tau \in \mathbb{R}$, $m \in \mathbb{N}$ and $1 \leq p < \infty$ define $$ L^p(\Omega, |x|^\tau) = \left\{u : \Omega \to \mathbb{R}: u \text{...
- 239
1 vote
1 answer
127 views
Comparing Mountain Pass level with Nehari manifold level
Consider $I : H^1_0(\Omega) \to \mathbb{R}$ defined by $I(u) = \frac{1}{2} \|u\|_{H^1_0(\Omega)}^2 - \frac{1}{p} \|u\|_{L^p(\Omega)}^p$, for $2 < p < 2^\ast$. It is well known that $$ I'(u)v = \...
- 239
1 vote
0 answers
64 views
Concluding a radial weak solution with radial test functions is a weak solution with all test functions
Let $B \subset \mathbb{R}^N$ be the unit ball and $E = \{u \in H^1_0(B) : u \text{ is radial}\}$. Define the functional $I : E \to \mathbb{R}$ by $$ I(u) = \int_{B} |\nabla u(x)|^2 dx - \frac{1}{p}\...
- 239
3 votes
1 answer
301 views
Understanding the Mountain Pass Theorem level
Consider: Mountain Pass Theorem: Let $X$ be a Banach space and $I \in C^1(X, \mathbb{R})$ such that $I(0) = 0$. Also suppose $G_1)$ There exists $\rho, \alpha > 0$ such that $I(u) \geq \alpha$ for ...
- 239
4 votes
0 answers
204 views
Elliptic PDE on Riemannian manifold with a gradient term $(\theta, d u)_g$ as the critical point of a functional
I wonder that could we write an elliptic PDE on Riemannian manifold with a gradient term $(\theta, d u)_g$ as the critical point of a functional, here $\theta$ is a $1-form$ and $(\theta, d u)_g$ is ...
- 1,011
4 votes
1 answer
213 views
Elliptic PDE question arising from a paper by Allard
I was reading Allard's 1986 paper "An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled" in Proc. ...
- 203
3 votes
1 answer
312 views
Question about divergence free vector fields and harmonic functions
Let $Z = (z_1,z_2,\dotsc, z_n)$ be a smooth vector field which is divergence free, i.e., $\operatorname{div}(Z) = 0$. I am trying to prove the following estimate: $$ \int_{B_1} \langle Z, \Delta Z \...
- 537
0 votes
0 answers
76 views
Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 537
0 votes
0 answers
46 views
Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 537
0 votes
0 answers
130 views
Curl-Div equation with singular matrix
I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
11 votes
0 answers
423 views
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution. In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13.8k
1 vote
0 answers
58 views
Uniform bound on the first moment for a perturbed advection-diffusion equation
I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line: $$ \begin{cases} u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x = ...
- 43
3 votes
0 answers
156 views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 464
1 vote
0 answers
62 views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$. By using the ...
- 617
3 votes
1 answer
301 views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation: $$ \begin{cases} \nabla \times (F\times\mathbf v)=g, \\ \operatorname{div}(\mathbf v)=0, \end{cases}\label{1}\tag{1} $$ where $F$ and $g$ are given vector fields. The ...
- 617
5 votes
0 answers
155 views
Fredholm index of degenerate elliptic PDE
We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$: $$ L(u):= -\operatorname{div}(a \, \nabla u) = 0, $$ where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
- 625
4 votes
0 answers
129 views
Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form $$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$ ...
- 103
6 votes
2 answers
310 views
Minimal assumptions for existence of solutions of First order PDE
I'm looking for a reference about existence of linear homogeneous first order PDE, in particular about the minimal assumption on the data. In literature I found that one require $C^1$-regularity on ...
- 81
9 votes
2 answers
779 views
Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
- 153
4 votes
1 answer
354 views
A certain solution for Sine-Gordon Equation
I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
- 247
5 votes
2 answers
424 views
Positivity for linear parabolic PDEs with time dependent coefficients
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
- 227
3 votes
1 answer
144 views
$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
I consider the following problem on the half real line $$ \begin{cases} u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm] u|...
- 43
0 votes
0 answers
110 views
Inside and up to boundary regularity improvement of linear differential operator
I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$ $$A=\sum a_\alpha\...
- 431
0 votes
0 answers
78 views
Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$): $$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
- 1,287
2 votes
0 answers
453 views
What is the fundamental solution for the backward heat equation?
According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
- 2,750
0 votes
0 answers
81 views
Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain
I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have \begin{equation}\label{...
- 431
3 votes
0 answers
173 views
Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$
Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
- 445
1 vote
0 answers
55 views
Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$: \begin{equation} u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1} \end{equation} Assume that $...
- 525
4 votes
3 answers
567 views
Generalized Fuchsian-type PDE
Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
- 149
0 votes
1 answer
307 views
About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
- 111
7 votes
2 answers
757 views
Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
user259525
2 votes
1 answer
283 views
Green's function for a linear PDE initial value problem
For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
- 1,538
4 votes
1 answer
219 views
Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$
I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ ...
- 161
0 votes
0 answers
149 views
Reversing heat transfer with respect to time
Fact: One can easily compute heat dispersion in a plane using the heat equation. Question: Has any research been done on computing the process in the reverse time direction? That is, given a heat map $...
- 119
1 vote
0 answers
241 views
Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 621
3 votes
0 answers
92 views
Convex combination of cyclically monotone sets
I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
- 105
1 vote
0 answers
185 views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1 vote
0 answers
47 views
Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO). Consider the following initial boundary value problem for the linear ...
- 4,294