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Results tagged with ap.analysis-of-pdes
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user 16183
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
7
votes
1
answer
354
views
Higher regularity of solutions of non-linear elliptic PDE
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Let $u\in C^2(\bar \Omega)$ be a solution of the Dirichlet problem for the non-linear equation
\begin{eqnarray}
F( …
- 21.8k
3
votes
1
answer
173
views
Solvability of general linear PDE with constant coefficients
Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any …
- 21.8k
2
votes
0
answers
45
views
Polynomial solutions of differential equations vs smooth ones
Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words
$$Im(D_1)\subset Ker …
- 21.8k
2
votes
0
answers
163
views
Inhomogeneous wave equation - a reference
Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\ …
- 21.8k
1
vote
0
answers
70
views
Solutions of constant coefficients differential operator on $\mathbb{R}^n$
Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier …
- 21.8k
1
vote
0
answers
88
views
Uniqueness of solution of linear PDE of first order
Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$
be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation
$$\partial_t \vec u(x,t)=\sum_{j=1} …
- 21.8k
2
votes
2
answers
731
views
Hölder estimates on solutions of non-linear elliptic PDE.
In his book "Some non-linear problems in Riemannian geometry" T.
Aubin states the following result (Theorem 3.56):
Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order
differential opera …
- 21.8k
2
votes
1
answer
260
views
A possible characterization of subharmonic functions
Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$ …
- 21.8k
2
votes
2
answers
611
views
Question on Sobolev spaces in domains with boundary
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$
$$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$
L …
- 21.8k
3
votes
3
answers
2k
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 …
- 21.8k
2
votes
1
answer
412
views
Second order estimates of Monge-Ampere equations
In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the …
- 21.8k
3
votes
0
answers
86
views
Estimate a function given an estimate of its Laplacian
Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions:
\begin{eqnarray*}
\int_B |f_\lambda(x)|^2dx\leq 1,\\ …
- 21.8k
1
vote
1
answer
701
views
Estimate on first derivatives given $L^2$-norm of Laplacian
Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions
$$X=\{u\in C^2(\bar B) \mid u|_{\partial B}=0 \text{ and } \|\Delta u\|_{L^2(B)}\leq 1\},$$
where $\Delta$ …
- 21.8k
2
votes
2
answers
141
views
Equality of spectra of products of operators
Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint.
Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities?
In my s …
- 21.8k
2
votes
0
answers
89
views
Second order estimates for Dirichlet problem for complex Monge-Ampere equation
Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -Ampe …
- 21.8k