The Dirichlet problem for a Petrovskiî elliptic system of second-order equations

1999 ◽  
Vol 40 (1) ◽  
pp. 195-203
Author(s):  
A. I. Yanushauskas
Keyword(s):  
Dirichlet Problem ◽  
Elliptic System ◽  
Second Order ◽  
Second Order Equations

On the Dirichlet Problem for a Second Order Elliptic System with Degeneration on the Entire Domain Boundary

1999 ◽  
Vol 6 (4) ◽  
pp. 395-400
Author(s):  
M. Usanetashvili
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic System ◽  
Domain Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Entire Domain ◽  
Order Elliptic System

Abstract The solvability of the first boundary value problem is investigated for a second order elliptic system with degeneration on the entire domain boundary.

scholarly journals Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

2021 ◽  
Author(s):  
Gioconda Moscariello ◽  
Giulio Pascale
Keyword(s):  
Dirichlet Problem ◽  
Unit Ball ◽  
Elliptic System ◽  
Positive Constant ◽  
Elliptic Systems ◽  
Second Order ◽  
Identity Matrix ◽  
Bmo Coefficients

AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.

The dirichlet problem for sub-elliptic second order equations

1997 ◽  
Vol 173 (1) ◽  
pp. 233-243 ◽  
Author(s):  
Alberto Parmeggiani ◽  
Chao-Jiang Xu
Keyword(s):  
Dirichlet Problem ◽  
Second Order ◽  
Second Order Equations

scholarly journals Positive solutions of some quasilinear singular second order equations

2004 ◽  
Vol 76 (1) ◽  
pp. 125-140 ◽  
Author(s):  
J. V. Goncalves ◽  
C. A. P. Santos
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Unit Ball ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Shooting Method ◽  
Second Order ◽  
Second Order Equations

AbstractIn this paper we study the existence and uniqueness of positive solutions of boundary vlue problems for continuous semilinear perturbations, say f: [0, 1) × (0, ∞) → (0, ∞), of class of quasilinear operators which represent, for instance, the radial form of the Dirichlet problem on the unit ball of RN for the operators: p-Laplacian (1 < p < ∞) ad k-Hessian (1 ≤ k ≤ N). As a key feature, f (r, u) is possibly singular at r = 1 or u =0, Our approach exploits fixed point arguments and the Shooting Method.

On the Dirichlet Problem for a Second Order Elliptic System with Degeneration on the Whole Boundary of a Multidimensional Domain

2008 ◽  
Vol 15 (4) ◽  
pp. 793-798
Author(s):  
Mikheil Usanetashvili
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic System ◽  
Boundary Value ◽  
Second Order ◽  
Entire Boundary ◽  
Order Elliptic System ◽  
Multidimensional Domain

Abstract The solvability of the first boundary value problem is studied for a second order elliptic system with degeneration on the entire boundary of a multidimensional domain.

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