Petronela Radu

Dirichlet’s principle and wellposedness of solutions for a nonlocal p-Laplacian system

Abstract We prove Dirichlet’s principle for a nonlocal p -Laplacian system which arises in the nonlocal setting of peridynamics when p = 2 . This nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain. Our analysis uses nonlocal versions of integration by parts techniques that resemble the classical Green and Gauss identities. The nonlocal energy functional associated with this “elliptic” type system exhibits a general kernel which could be weakly singular. The coercivity of the system is shown by employing a nonlocal Poincare’s inequality. We use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy, from which we obtain the wellposedness of this steady state diffusion system.

Decay estimates for wave equations with variable coefficients

We establish weighted L 2 -estimates for dissipative wave equations with variable coefficients that exhibit a dissipative term with a space dependent potential. These results yield decay estimates for the energy and the L 2 —norm of solutions. The proof is based on the multiplier method where multipliers are specially engineered from asymptotic profiles of related parabolic equations.

THE GENERALIZED DIFFUSION PHENOMENON AND APPLICATIONS

We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainders. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.

Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling

In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying

A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

L_\infty

Strong solutions for semilinear wave equations with damping and source terms

-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.

Differentiability and integrability properties for solutions to nonlocal equations

We consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in (Math Models Methods Appl Sci 23(03):493–540, 2013) and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary-value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their L2-strong convergence to functions in W2,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary-value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.

Taking Math Outside of the Classroom:Math in the City

This article deals with local existence of strong solutions for semilinear wave equations with power-like interior damping and source terms. A long-standing restriction on the range of exponents for the two nonlinearities governs the literature on wellposedness of weak solutions of finite energy. We show that this restriction may be eliminated for the existence of higher regularity solutions by employing natural methods that use the physics of the problem. This approach applies to the Cauchy problem posed on the entire ℝ n as well as for initial boundary problems with homogeneous Dirichlet boundary conditions.

Bridging Local and Nonlocal Models: Convergence and Regularity

The main results of this paper concern integrability and differentiability properties for solutions for a nonlocal system. The boundary value problem is associated with an integro-differential equation that exhibits a weakly singular kernel. The main result provides a nonlocal equivalent of classical regularity theorems established for elliptic systems. In addition, we show that solutions inherit additional integrability properties of the forcing term, thus providing a simple proof to well-posedness of solutions in Lp for p > 2.

Diffusion phenomenon in Hilbert spaces and applications

Abstract Math in the City is an interdisciplinary mathematics course offered at University of Nebraska-Lincoln in which students engage in a real-world experience to understand current major societal issues of local and national interest. The course is run in collaboration with local businesses, research centers, and government organizations, that provide data and act as consultants throughout the course. We provide a brief description of the course with examples of projects offered over the years. We emphasize service-learning aspects of the program, as well as benefits to students and business collaborators.