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5 votes
0 answers
879 views

A fourth-order linear PDE

I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $$x^3 f_{xxxt}+ f =0$$ Does anyone know if this type of PDE already appeared in the literature? ...
4 votes
3 answers
473 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 $t$...
6 votes
2 answers
448 views

About the index theorems

I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
2 votes
1 answer
215 views

Reference request for spectral theory of elliptic operators [closed]

I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference. I ...
2 votes
1 answer
328 views

The study of dynamics of a polynomial vector field via Green's function methods

In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...
4 votes
0 answers
110 views

Regularity of the solution to a differential system with variable coefficients

Let $\Omega\subset \mathbb R^n$ be a convex subset. All the objects below will be defined on this set. Let us assume $P(x,D)$ to be a differentiable operator order $m$ and of square size, that is ...
1 vote
1 answer
442 views

Cauchy problem for an overdetermined system of PDE

This question is strictly related to this one. Let us consider the differential system with constant coefficients $$\left(\begin{array}{ccc} B_{11} & B_{12} & 0\\ ...
4 votes
3 answers
2k views

book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...