Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

“…We will explain possible applications in detail for the biharmonic Alt-Caffarelli problem, which has recently raised a lot of interest, cf. [8], [7], [23]. Some methods, e.g the blow-up techniques we use, are also inspired by applications from free boundary problems [1, Section 4], but need to be refined in our analysis.…”

Section: Introductionmentioning

confidence: 99%

The Poisson equation involving surface measures

Müller¹

2021

Preprint

View full text Add to dashboard Cite

We prove the (optimal) W 1,8 -regularity of weak solutions to the equation ´∆u " Q H n´1 Γ in a domain Ω Ă R n with Dirichlet boundary conditions, where Γ ĂĂ Ω is a compact (Lipschitz) manifold and Q P L 8 pΓq.We also discuss optimality and necessity of the assumptions on Q and Γ.Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.

show abstract

“…A related model is the singularly perturbed bi-Laplace equation ∆ 2 u = −β (u ), which can be thought of as the biharmonic counterpart of the classical combustion problem. This problem was investigated by Dipierro, Karakhanyan, and Valdinoci in [15], where they proved the convergence as → 0 to a free boundary problem driven by the bi-Laplacian. Moreover, they derived a monotonicity formula in the plane, and used it to establish the quadratic behavior of solutions near the zero level set.…”

Section: Introductionmentioning

confidence: 99%

A survey on obstacle-type problems for fourth order elliptic operators

Danielli¹,

Ali²

2022

View full text Add to dashboard Cite

In this article we give a brief overview of some known results in the theory of obstacle-type problems associated with a class of fourth-order elliptic operators, and we highlight our recent work with collaborators in this direction. Obstacle-type problems governed by operators of fourth order naturally arise in the linearized Kirchhoff-Love theory for plate bending phenomena. Moreover, as first observed by Yang in [34], boundary obstacle-type problems associated with the weighted bi-Laplace operator can be seen as extension problems, in the spirit of the one introduced by Caffarelli-Silvestre, for the fractional Laplacian (−∆) s in the case 1 < s < 2. In our recent work, we investigate some problems of this type, where we are concerned with the well-posedness of the problem, the regularity of solutions, and the structure of the free boundary. In our approach, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas.

show abstract

Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

Cited by 9 publications

References 19 publications

Polyharmonic equations involving surface measures

Polyharmonic equations involving surface measures

The Poisson equation involving surface measures

A survey on obstacle-type problems for fourth order elliptic operators

Contact Info

Product

Resources

About