Well-posedness and approximation of solutions of linear divergence-form elliptic problems on exterior regions

2014 ◽  
Vol 38 (9) ◽  
pp. 1867-1875 ◽  
Author(s):  
Giles Auchmuty ◽  
Qi Han
Keyword(s):  
Elliptic Problems ◽  
Divergence Form ◽  
Well Posedness ◽  
Exterior Regions

scholarly journals Pathological solutions to elliptic problems in divergence form with continuous coefficients

2009 ◽  
Vol 347 (13-14) ◽  
pp. 773-778 ◽  
Author(s):  
Tianling Jin ◽  
Vladimir Maz'ya ◽  
Jean Van Schaftingen
Keyword(s):  
Elliptic Problems ◽  
Divergence Form

scholarly journals Explicit Estimates for Solutions of Mixed Elliptic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Potential Theory ◽  
Mixed Problem ◽  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Order Equation ◽  
Second Order ◽  
Green Functions ◽  
Mixed Problems ◽  
Quantitative Estimates

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

A note on positivity of two-dimensional differential operators

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4651-4663 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Sema Akturk
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Half Plane ◽  
Differential Operators ◽  
Elliptic Problems ◽  
Well Posedness ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
Fractional Spaces ◽  
Uniform Ellipticity

We consider the two-dimensional differential operator A(t,x)u(t,x) = -a11 (t, x) utt -a22(t,x)uxx +?u defined on functions on the half-plane R+ x R with the boundary condition u(0,x) = 0, x ? R where aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a2 11(t,x) + a222(t,x)? ? > 0, ? > 0. The structure of fractional spaces E?,1 (L1 (R+ x R), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2?1(R+ x R)) spaces is established. In applications, theorems on well-posedness in L1 (W2?1 (R+ x R)) spaces of elliptic problems are obtained.

scholarly journals Uniqueness in MHD in divergence form: Right nullvectors and well-posedness

2002 ◽  
Vol 43 (12) ◽  
pp. 6195-6208 ◽  
Author(s):  
Maurice H. P. M. van Putten
Keyword(s):  
Divergence Form ◽  
Well Posedness

scholarly journals The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

2019 ◽  
Vol 29 (10) ◽  
pp. 1819-1851 ◽  
Author(s):  
K. Disser ◽  
J. Rehberg
Keyword(s):  
Charge Carrier ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Divergence Form ◽  
Well Posedness ◽  
Elliptic Regularity ◽  
Surfaces And Interfaces ◽  
Interfacial Recombination ◽  
Semiconductor Equations ◽  
Three Space

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

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