Search results for "continuity"
showing 10 items of 378 documents
Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian
2018
We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Rad\'o-type removability theorem.
Gradient and Lipschitz Estimates for Tug-of-War Type Games
2021
We define a random step size tug-of-war game and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in the higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as an improved version of the cylinder walk argument. peerReviewed
An evolutionary Haar-Rado type theorem
2021
AbstractIn this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.
Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities
2018
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in $\Omega\times \Omega$ via couplings.
On the regularity of very weak solutions for linear elliptic equations in divergence form
2020
AbstractIn this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .
Gravity-derived Moho map for Latvia
2020
A precise understanding of crustal structure is essential to the fields of geodynamics, seismology and certain branches of geophysics. A boundary between the crust and the mantle is known as the MohoroviÄiÄ discontinuity, simply referred to as the âMohoâ. Moho geometry and depth have been extensively studied in Europe, but there are still regions with little information about it. One such area is the northern Baltics, Latvia in particular. So far, only one seismic refraction profile, spanning from Sovetsk (Kaliningrad) to Kohtla-Järve (Estonia), has been used to study the deep structure of the Earth in Latvia. We propose gravity inversion (ParkerâOldenburg algorithm) to gain more insight i…
A generalized methodology for distribution systems faults identification, location and characterization
2005
Service continuity is of basic importance in the definition of the quality of the electrical energy, for this reason, the research in the field of faults diagnostic for distribution systems is spreading ever more. In this paper, a new methodology for diagnostic management of automated distribution systems is presented. The technique is based on the solution of a circuital model of the electrical system resulting from the composition of distributed parameters quadripoles. The solution gives as a result the identification of the type of fault, of its characteristic parameters and location. The paper shows an application to line to line grounded and ungrounded faults in which also its precisio…
Higher-order energy density functionals in nuclear self-consistent theory
2011
In this thesis consisting of two publications and an overview part, a study of two aspects of energy density functionals has been performed. Firstly, we have linked the next-to-next-to-next-to-leading order nuclear energy density functional to a zero-range pseudopotential that includes all possible terms up to sixth order in derivatives. Within the Hartree-Fock approximation, the quasi-local nuclear Energy Density Functional (EDF) has been calculated as the average energy obtained from the pseudopotential. The direct reference of the EDF to the pseudopotential acts as a constraint that allows for expressing the isovector coupling constants functional in terms of the isoscalar ones, or vice …
Multidimensional P-adic Integrals in some Problems of Harmonic Analysis
2017
The paper is a survey of results related to the problem of recovering the coefficients of some classical orthogonal series from their sums by generalized Fourier formulas. The method is based on reducing the coefficient problem to the one of recovering a function from its derivative with respect to an appropriate derivation basis. In the case of the multiple Vilenkin system the problem is solved by using a multidimensional P-adic integral.
Mappings of L p -integrable distortion: regularity of the inverse
2016
Let X be an open set in R n and suppose that f : X → R n is a Sobolev homeomorphism. We study the regularity of f −1 under the L p -integrability assumption on the distortion function Kf . First, if X is the unit ball and p > n−1, then the optimal local modulus of continuity of f −1 is attained by a radially symmetric mapping. We show that this is not the case when p 6 n − 1 and n > 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for |Df −1 | in terms of the L p -integrability assumptions of Kf . peerReviewed