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Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

100%

Pliś M. E., Ziemian B.

Annales Polonici Mathematici

1997

tom 67

nr 1

31-41

We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.

A counterexample to the $L^{p}$-Hodge decomposition

100%

Hajłasz P.

Banach Center Publications

1996

tom 33

nr 1

79-83

We construct a bounded domain $Ω ⊂ ℝ^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}(Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^{p}(Ω,ℝ^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}(Ω)$ for all p > 4.

Infinite dimension of solutions of the Dirichlet problem

100%

Ryazanov V.

Open Mathematics

2015

tom 13

nr 1

It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Banach Center Publications

1 Open Mathematics

1 Hajłasz P.

1 Pliś M. E.

1 Ryazanov V.

1 Ziemian B.

1 2015

1 1997

1 1996

Wyniki wyszukiwania

Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

A counterexample to the $L^{p}$-Hodge decomposition

Infinite dimension of solutions of the Dirichlet problem