Direct Methods in the Theory of Elliptic Equations

Nečas, Jindřich; Nečasová, Šárka; Simader, Christian G.

doi:10.1007/978-3-642-10455-8

Cited by 356 publications

(272 citation statements)

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“…(for a proof of Green's formula see page 20 [29]). This equality is a consequence of the divergence Theorem, (for a proof of the divergence Theorem stated by E. Heinz see page 46 [36], also page 570 [11]).…”

Section: Black Hole Name Solar Mass (Sun=1) Referencesmentioning

confidence: 99%

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Spacetime singularity, singular bounds and compactness for solutions of the Poisson's equation

Aranda¹

2015

Cubo

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A black hole is a spacetime region in whose interior lies a structure known as a spacetime singularity whose scientific description is profoundly elusive, and which depends upon the still missing theory of quantum gravity. Using the classical weak comparison principle we are able to obtain new bounds, compactness results and concentration phenomena in the theory of Newtonian potentials of distributions with compact support which gives a suitable mathematical theory of spacetime singularity. We derive a rigorous renormalization of the Newtonian gravity law using nonlinear functional analysis and we have a solid set of astronomical observations supporting our new equation. This general setting introduces a new kind of ill posed problem with a very simple physical interpretation. RESUMENUn hoyo negro es una región espacio-temporal en cuyo interior hay una estructura llamada singularidad espacio-temporal cuya descripción científica es difícil de encontrar, y que depende de la aún inexistente teoría de la gravedad cuántica. Usando el clásico principio de comparación débil, aquí probamos nuevas cotas, resultados de compacidad y fenómenos de concentración en la teoría de potenciales Newtonianos de distribuciones de soporte compacto, que dan una teoría matemática adecuada de la singularidad espacio-temporal. Derivamos una rigurosa renormalización de la ley de gravitación Newtoniana usando análisis funcional no lineal y tenemos un contundente conjunto de datos de observaciones astronómicas que apoyan nuestra nueva ecuación. Este marco general introduce una nueva forma de problema mal-puesto con una interpretación física muy simple.

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Section: Black Hole Name Solar Mass (Sun=1) Referencesmentioning

confidence: 99%

“…We point that C ∞ extension is a nontrivial task, see for example page 136 [18] or the Nikolskii's extension method page 69 [29] and the Calderon's extension method page 72 [29]. Therefore we give a full description of our procedure.…”

Section: Moreovermentioning

confidence: 99%

Spacetime singularity, singular bounds and compactness for solutions of the Poisson's equation

Aranda¹

2015

Cubo

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“…Applying classical PDE theory, in particular the Lax-Friedrich inequality and Grönwall's lemma (for more details, see [12,17]), gives:…”

Section: Bound On Pricing Errormentioning

confidence: 99%

“…where α is some positive constant, related to the V −ellipticity of the form Ω eL 1 edΩ [17]. We use the notation U := (L 2 − L 1 ) C and find the following operator:…”

Section: Bound On Pricing Errormentioning

confidence: 99%

The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation

Stoep¹,

Grzelak

Oosterlee

2013

SSRN Journal

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In this article we propose an efficient Monte Carlo scheme for simulating the stochastic volatility model of Heston [14] enhanced by a non-parametric local volatility component. This hybrid model combines the main advantages of the Heston model and the local volatility model introduced by Dupire [8] and Derman & Kani [7]. In particular, the additional local volatility component acts as a "compensator" that bridges the mismatch between the non-perfectly calibrated Heston model and the market quotes for European-type options. By means of numerical experiments we show that our scheme enables a consistent and fast pricing of products that are sensitive to the forward volatility skew. Detailed error analysis is also provided.

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“…Weighted Sobolev spaces related to inequalities with various weights are investigated e.g. in books [51,63] and papers [48,49,64]. Furthermore, Orlicz-Sobolev spaces are considered e.g.…”

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confidence: 99%

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

Skrzypczak

2014

Nonlinear Differ. Equ. Appl.

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Abstract. We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving ALaplacian −ΔAu = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω ⊆ R n . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the formwhereĀ(t) is a Young function related to A and satisfying Δ -condition, while FĀ(t) = 1/(Ā(1/t)). Examples involvingĀ(t) = t p log α (2+t), p ≥ 1, α ≥ 0 are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30-50, 2013), where we dealt with inequality −Δpu ≥ Φ, leading to Hardy and Hardy-Poincaré inequalities with the best constants. Mathematics Subject Classification (2010). Primary 26D10; Secondary 31B05, 35D30, 35J60, 35R45.

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Direct Methods in the Theory of Elliptic Equations

Cited by 356 publications

References 0 publications

Spacetime singularity, singular bounds and compactness for solutions of the Poisson's equation

Spacetime singularity, singular bounds and compactness for solutions of the Poisson's equation

The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

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