# Tagged Questions

Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

44 views

### The composition of a dissipative operator and a positive operator is dissipative?

Consider the following bilinear system on a open and bounded domain $\Omega$ \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &...
34 views

### $L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation: u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ ...
31 views

149 views

### Recover Embedding from Metric

Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known. And suppose that I know ...
73 views

### Imposing boundary conditions and self-similarity on a PDE

This question is an exact duplicate of the question Imposing boundary conditions AND self-similarity on a PDE posted by Stan Corey Carter on math.stackexchange.com. I have a PDE in the ...
45 views

### Conservation of charge and energy in the Schrödinger equation

In Cazenave's Semilinear Schrödinger Equation, page 56, he describes derivation of conservation of charge and energy of the equation $iu_t+\Delta u+|u|u=0$, ($\alpha=\lambda=1$ and $n=3$, if referring ...
59 views

It shall be an old story in PDE. I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of $$\inf_{a \in [-1,1]} \{... 0answers 59 views ### A priori C^0 estimates for a semi-linear vector Poisson equation Main Question Consider a C^2,H^2 map F:\mathbb{R}^m \to \mathbb{C}^n which satisfies the following equation:$$ -\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$Here a_i:... 0answers 31 views ### The jump set of SBV function for different value of parameter in image denoising problem The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each \alpha>0 where \Omega\subset \mathbb R^2 is open bounded with sommth boundary,$$ u_\...
Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...