# Tagged Questions

28 views

### Reference for the following equation

Can someone suggest me references about the following equation $$u_t+A\cdot\nabla u=i\Delta u$$ with $A$ a smooth vector field.
30 views

### $C^0$ estimate for solutions of the Neumann problem

I am interested in a reference for (or counterexample to!) a particular $C^0$ estimate for solutions of the Laplace equation with Neumann boundary conditions. More precisely, let $(M,g)$ be a smooth, ...
57 views

31 views

### Studying Hamiltonian PDEs - Where to start?

I first got in contact with Hamiltonian PDEs when I wrote my first thesis about the Nash-Moser-Theorem as some books mentioned them as a field of application of this theorem. As those examples were ...
34 views

### Regularity for a parabolic problem with nonsmooth coefficients

I'm looking for references on the regularity of the (weak) solution to the parabolic problem with nonsmooth coefficients. In most literature, like Evans, the coefficients are often assumed to be ...
15 views

60 views

### Elliptic PDEs in Banach space

The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point ...
26 views

### Existence and regularity of 2D stream function.

Suppose $\Omega$ is a bounded open simply connected region in $\mathbb{R}^2$, and let $u:\overline{\Omega} \to \mathbb{R}^2$. Suppose $u$ is divergence free. Then there exists a stream function $\psi$ ...
19 views

168 views

### What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
13 views

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ... 0answers 31 views ### Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator. I am considering$L\phi = -\triangle \phi + u \cdot \nabla \phi$and its "adjoint" operator$L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$on a bounded domain$\Omega \subseteq \mathbb{R}^n$. ... 1answer 52 views ### What are good resources for learning Numerical methods for Partial Differential Equations? I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a ... 0answers 36 views ### Elliptic regularity of Dirichlet problem Suppose$\Omega\subset\mathbb{C}$is a simply connected domain with$C^\infty$boundary. Consider the following Dirichlet problem $$\Delta u |_{\Omega}=0$$ $$u|_{\partial\Omega}=f$$ Under what ... 0answers 44 views ### Regularity of Dirichlet Eigenvalues on Lipschitz Domain What kind of regularity do we generally have for weak solutions to the Dirichlet problem? $$(\Delta+\lambda)u=0 \textrm{ in }U$$ $$u=0 \textrm{ on }\partial U$$ where$U$is a planar domain with ... 0answers 23 views ### Conjugating an operator with a gauge transformation; how is the kernel affected. For the differential operator $$D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m)$$ where$A(x)$is Hermitian and$C^\infty_T ...
I'm reading a paper and the authors applied the following sobolev type estimates $$||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}$$ for $\alpha>\frac{1}{4}$, where $v$ ...
I am trying to understand the classification of second order PDE's from this article. In page 45, line 1 can somebody please explain to me how $\partial^2U \over {\partial x_i\partial x_j}$ was ...