The Method of Potential Operators for Anisotropic Helmholtz Operators on Domains with Smooth Unbounded Boundaries

2017 ◽  
pp. 229-256 ◽  
Author(s):  
Vladimir Rabinovich
Keyword(s):  
Potential Operators ◽  
Helmholtz Operators

scholarly journals Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

2020 ◽  
Vol 23 (2) ◽  
pp. 378-389
Author(s):  
Ferenc Izsák ◽  
Gábor Maros
Keyword(s):  
Fractional Order ◽  
Boundary Data ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Integral Form ◽  
Uniqueness Result ◽  
Boundary Integral ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Potential Operators

AbstractFractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. The existence-uniqueness result is stated also for two-dimensional domains. Finally, a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.

scholarly journals Potential Operators on Cones of Nonincreasing Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Alexander Meskhi ◽  
Ghulam Murtaza
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Product Type ◽  
Necessary And Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Potential Operators ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators(Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dtand(ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτon the cone of nonincreasing functions are derived. In the case ofℐα1,α2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.

scholarly journals Ensemble Generalized Kohn-Sham Theory: The Good, the Bad, and the Ugly

2020 ◽  
Author(s):  
Tim Gould ◽  
Leeor Kronik
Keyword(s):  
Potential Theory ◽  
Excited States ◽  
Effective Potential ◽  
Open Systems ◽  
Potential Operators ◽  
Optimized Effective Potential ◽  
Quantum Ensembles

Two important extensions of Kohn-Sham (KS) theory are generalized KS theory and ensemble KS theory. The former allows for non-multiplicative potential operators and greatly facilitates practical calculations with advanced, orbital-dependent functionals. The latter allows for quantum ensembles and enables the treatment of, e.g., open systems and excited states. Here, we combine the two extensions, both formally and practically, first via an exact yet complicated formalism, then via a computationally tractable variant that involves a controlled approximation of ensemble "ghost interactions" by means of an approach inspired by optimized effective potential theory. The resulting formalism is illustrated using selected examples. This opens the door to the application of generalized KS theory in more challenging quantum scenarios and to the improvement of ensemble theories for the purpose of practical and accurate calculations.<br>

scholarly journals A saddle point type solution for a system of operator equations

2021 ◽  
pp. 1-12
Author(s):  
Piotr Kowalski
Keyword(s):  
Saddle Point ◽  
Minimax Theorem ◽  
Dirichlet Problems ◽  
Type Solution ◽  
Operator Equations ◽  
Potential Operators ◽  
System Of Operator Equations ◽  
Ky Fan ◽  
Point Type

Let Ω⊂Rn n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where N:R×R→R is C1 and is partially convex-concave and A:W01,p(Ω)→(W01,p(Ω))* B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

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