Search results for "Laplacian"
showing 10 items of 135 documents
Calderón problem for the p-Laplace equation : First order derivative of conductivity on the boundary
2016
We recover the gradient of a scalar conductivity defined on a smooth bounded open set in Rd from the Dirichlet to Neumann map arising from the p-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when p 6 = 2. In the p = 2 case boundary determination plays a role in several methods for recovering the conductivity in the interior. peerReviewed
A model of capillary phenomena in RN with subcritical growth
2020
This paper deals with the nonlinear Dirichlet problem of capillary phenomena involving an equation driven by the p-Laplacian-like di¤erential operator in RN. We prove the existence of at least one nontrivial nonnegative weak solution, when the reaction term satisfies a sub-critical growth condition and the potential term has certain regularities. We apply the energy functional method and weaker compactness conditions.
Fractional p-Laplacian evolution equations
2016
Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for x ∈ Ω , t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problem…
A tour of the theory of absolutely minimizing functions
2004
A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation
Singular solutions to p-Laplacian type equations
1999
We construct singular solutions to equations $div\mathcal{A}(x,\nabla u) = 0,$ similar to the p-Laplacian, that tend to ∞ on a given closed set of p-capacity zero. Moreover, we show that every Gδ-set of vanishing p-capacity is the infinity set of some A-superharmonic function.
Variational differential inclusions without ellipticity condition
2020
The paper sets forth a new type of variational problem without any ellipticity or monotonicity condition. A prototype is a differential inclusion whose driving operator is the competing weighted $(p,q)$-Laplacian $-\Delta_p u+\mu\Delta_q u$ with $\mu\in \mathbb{R}$. Local and nonlocal boundary value problems fitting into this nonstandard setting are examined.
Multiple solutions for a discrete boundary value problem involving the p-Laplacian.
2008
Multiple solutions for a discrete boundary value problem involving the p-Laplacian are established. Our approach is based on critical point theory.
TUG-OF-WAR, MARKET MANIPULATION, AND OPTION PRICING
2014
We develop an option pricing model based on a tug-of-war game involving the the issuer and holder of the option. This two-player zero-sum stochastic differential game is formulated in a multi-dimensional financial market and the agents try, respectively, to manipulate/control the drift and the volatility of the asset processes in order to minimize and maximize the expected discounted pay-off defined at the terminal date $T$. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a partial differential equation involving the non-linear and completely degenerate parabolic infinity Laplace operator.
Local regularity for time-dependent tug-of-war games with varying probabilities
2016
We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain H\"older and Harnack estimates. The games have a connection to the normalized $p(x,t)$-parabolic equation $(n+p(x,t))u_t=\Delta u+(p(x,t)-2) \Delta_{\infty}^N u$.
A Parametric Dirichlet Problem for Systems of Quasilinear Elliptic Equations With Gradient Dependence
2016
The aim of this article is to study the Dirichlet boundary value problem for systems of equations involving the (pi, qi) -Laplacian operators and parameters μi≥0 (i = 1,2) in the principal part. Another main point is that the nonlinearities in the reaction terms are allowed to depend on both the solution and its gradient. We prove results ensuring existence, uniqueness, and asymptotic behavior with respect to the parameters.