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.
2,819
questions
2
votes
0answers
16 views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
3
votes
0answers
39 views
Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries
Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...
1
vote
0answers
110 views
Minimal assumption for an elliptic equation
On the disc $\mathbb{D}$ on the disc with a metric $g=e^{2\lambda} \vert dz \vert^2$(let assume $\lambda$ is smooth on $\overline{\mathbb{D}}$) and I consider either
$$\newcommand{\Div}{\operatorname{...
5
votes
0answers
55 views
A bounded extension operator
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
-2
votes
0answers
37 views
Well-posedness in modified H2 space
Can I modify the $H^2$ space such that:
$$\tilde{H}^2 := \left[ u(\Omega): ||u||^2_{L_2(\Omega)} + ||\nabla u||^2_{L_2(\Omega)} + ||\Delta u||_{L_2(\Omega)}^2 < \infty \right]$$
and then use the ...
3
votes
0answers
102 views
Reference on noncommutative PDE
I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
0
votes
0answers
50 views
Trace inequality normal derivative
For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality:
$$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$
which can be found in many places in the ...
5
votes
0answers
104 views
An inequality for the Hessian of eigenfunctions of the Laplacian on compact manifolds
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator. Let $\lambda_1 >0$ be the first positive eigenvalue. That is, there exists a non-trivial function ...
2
votes
0answers
53 views
Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator
I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why ...
1
vote
1answer
36 views
Computing the fractional Laplacian of power function
Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?
4
votes
1answer
117 views
Reduce PDE to ODE by dilation symmetry
I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753.
Consider the following PDE: ...
2
votes
1answer
74 views
Extension of outer unit normal vector to interior
Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
1
vote
3answers
176 views
Uniqueness of solution of the wave equation
Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$
with initial conditions
$$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$
Does ...
1
vote
1answer
124 views
Is there any nontrivial characterization of weakly differentiable functions?
When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...
0
votes
0answers
33 views
A functional that occurs in Vlasov-Poisson equation
Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
1
vote
1answer
60 views
fractional Laplacian estimates
Suppose $(-\Delta)^s u=f \geq 0$ in a ball $B_2$ and $u=0$ in $ R^N \setminus B_2.$ Also suppose $u$ is $C^{s}$ non-negative and $(-\Delta)^s u=0$ in $B_2 \setminus B_1$ and $u\leq a$ on $\partial B_1$...
2
votes
0answers
59 views
Inhomogeneous wave equation - a reference
Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
-1
votes
0answers
39 views
Understanding weak formulation of a linear elliptic pde [closed]
hey I'm working on weak formulation on an elliptic pde and I have these questions:
Is there any difference between: $\Delta u=f$ in $\Omega$ and $\Delta u=f$ almost everywhere in $\Omega$ when $\...
3
votes
1answer
94 views
Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
1
vote
1answer
59 views
Poincaré-type Inequality
In Lieb's paper "On the lowest eigenvalue of the Laplacian
for the intersection of two domains" one finds the following remark:
Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
4
votes
1answer
115 views
Decay of Fourier coefficients of real analytic functions
I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, ...
2
votes
1answer
83 views
What is the example of non-regular boundary point?
I am studying in PDE and I have next definition :
Definition. Let $\Omega\subset\mathbb{R}^n$ open, connected. Then $\xi\in\partial\Omega$ is regular if there exists a superharmonic function $p$ in ...
0
votes
1answer
68 views
A time dependent variational problem coming from a second order linear PDE
Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...
0
votes
1answer
119 views
Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?
Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...
1
vote
1answer
57 views
Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
6
votes
1answer
383 views
Solutions of PDE under changing topology
Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we ...
2
votes
1answer
67 views
On $s$-harmonic functions
Is this statement true?
A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...
1
vote
0answers
96 views
Existence theory with an integral equation
I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system ...
6
votes
0answers
269 views
A question in elementary differential geometry
Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...
2
votes
2answers
102 views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
0
votes
1answer
124 views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
4
votes
0answers
160 views
Uniqueness condition for Hamilton-Jacobi equation?
Let $f= f(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function such that
$$
\partial_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R}...
3
votes
1answer
83 views
Space of holomorphic functions multiplied by smooth functions taking real values
Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
1
vote
0answers
45 views
Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
0
votes
0answers
36 views
A nonlinear PDE on a matrix domain involving a maximum eigenvalue operator
In a research problem in optimal control
$^{\ast}$ I came across the following nonlinear first-order PDE:
$$\frac{\partial V}{\partial t}=\max\text{eig}\left[\sum_{i=1}^m P_i\frac{\partial V}{\partial ...
9
votes
2answers
1k views
Can learning Riemann surfaces be more beneficial than numerical analysis for an analyst?
I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, ...
1
vote
0answers
91 views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
3
votes
0answers
76 views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
2
votes
1answer
292 views
Is there a diffeomorphism of the disk with constant sum of singular values?
This question is a relaxed version of this question.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$.
Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...
4
votes
1answer
119 views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
2
votes
0answers
41 views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
0
votes
0answers
35 views
When does the solution to the Fokker-Planck equation admit a density wrt Lebesgue measure?
Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation
$$\frac{\partial u(t,x)}{\partial t}= \...
1
vote
0answers
68 views
Strichartz estimate for the Schrödinger equation
Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t ...
2
votes
0answers
76 views
Wave equation for smooth Schwartz kernels
Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...
2
votes
1answer
69 views
Approximating functions in $H^1_0(U) \cap H^2(U)$ via $H^1$ norm and $L^2$ projection
Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \...
3
votes
0answers
60 views
Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
7
votes
0answers
150 views
Smoothness of solution map for PDE
I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
1
vote
0answers
60 views
Solution existence for two-dimensional parabolic PDEs
I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
1
vote
0answers
32 views
Are there maximum principles related to the third boundary condition?
I am working on this problem
\begin{equation}
\begin{cases}
u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\
u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0,
\end{cases}
\end{...
5
votes
1answer
128 views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
