$H^2$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $\pi$

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

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Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2041 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Víctor Hernández-Santamaría ◽

Alberto Saldaña

Keyword(s):

Critical Exponent ◽

Convergence Properties ◽

Scaling Invariance ◽

Convergence Of Solutions ◽

Nonexistence Result ◽

Homogeneous Sobolev Space ◽

Similar Characterization ◽

Nonnegative Solutions ◽

Least Energy ◽

Local Transition

Abstract We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ⁢ ( Ω ) {D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. u s {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t > 0 {t>0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {t-\varepsilon} . A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s > 1 {s>1} .

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Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v32i2.17522 ◽

2014 ◽

Vol 32 (2) ◽

pp. 83 ◽

Cited By ~ 1

Author(s):

Mohammed Massar ◽

EL Miloud Hssini ◽

Najib Tsouli

Keyword(s):

Saddle Point ◽

Point Theorem ◽

Elliptic Problem ◽

Weak Solutions ◽

Mountain Pass Theorem ◽

Mountain Pass ◽

Saddle Point Theorem ◽

Biharmonic Problem ◽

Existence And Multiplicity ◽

Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

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Exponential Decay for a System of Equations with Distributed Delays

Journal of Applied Mathematics ◽

10.1155/2015/981383 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6

Author(s):

Nasser-Eddine Tatar

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Exponential Decay ◽

Distributed Delay ◽

Distributed Delays ◽

System Of Equations ◽

Activation Functions ◽

Convergence Of Solutions

We prove convergence of solutions to zero in an exponential manner for a system of ordinary differential equations. The feature of this work is that it deals with nonlinear non-Lipschitz and unbounded distributed delay terms involving non-Lipschitz and unbounded activation functions.

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Season 3: Product Governance. Rethinking Retail Customer Protection in the EU Insurance Market

Global Jurist ◽

10.1515/gj-2017-0016 ◽

2017 ◽

Vol 18 (1) ◽

Author(s):

Lydia Velliscig

Keyword(s):

Insurance Market ◽

Business Rules ◽

Best Interest ◽

Convergence Of Solutions ◽

New Frontier ◽

Disclosure Of Information ◽

The Law ◽

The Eu

AbstractThe regulation of all traditional branches - banking, securities and insurance - of financial law is going through a change in retail customer protection: in this area of the law, an eventual convergence of solutions in client protection initiatives may be found. In a context oriented towards acting in accordance with the best interest of customers, EU legislator currently seeks a new “frontier” in the protection of retail customers and tends to develop new tools and strategies in addition to the disclosure of information and conduct of business rules, in order to remove potentially detrimental products from the market. This contribution examines the “product oversight and governance” principle intended to remedy problems associated with products misselling. In the details, this trend is analyzed with reference to the upcoming insurance distribution directive.

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