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- Kim, Sunghan, et al.
(författare)
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An elliptic free boundary arising from the jump of conductivity
- 2017
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Ingår i: Nonlinear Analysis. - : Elsevier Ltd. - 0362-546X .- 1873-5215. ; 161, s. 1-29
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider a quasilinear elliptic PDE, div(A(x,u)∇u)=0, where the underlying physical problem gives rise to a jump for the conductivity A(x,u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.
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| 5. |
- Kim, Sunghan, et al.
(författare)
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Homogenization of a Singular Perturbation Problem
- 2019
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Ingår i: Journal of Mathematical Sciences. - : Springer-Verlag New York. - 1072-3374 .- 1573-8795. ; 242:1, s. 163-176
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Tidskriftsartikel (refereegranskat)abstract
- We discuss homogenization of the singular perturbation problem Δpuδε=fεβδ(uδε)inℝn\B1¯ with a constant boundary value on the ball. Here, Δp is the usual p-Laplacian operator. It is generally understood that the two parameters δ and ε are in competition and two different behaviors may be exhibited, depending on which parameter tends to zero faster. We consider one scenario where we assume that ε, the homogenization parameter, tends to zero faster than δ, the singular perturbation parameter. We show that there is a universal speed for which the limit solves a standard Bernoulli free boundary problem.
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- Kim, Sunghan, et al.
(författare)
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HOMOGENIZATION OF THE BOUNDARY VALUE FOR THE DIRICHLET PROBLEM
- 2019
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Ingår i: Discrete and Continuous Dynamical Systems. - : AMER INST MATHEMATICAL SCIENCES-AIMS. - 1078-0947 .- 1553-5231. ; 39:12, s. 6843-6864
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in C-1-domains.
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- Kim, Sunghan, et al.
(författare)
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Nodal Sets for "Broken" Quasilinear PDEs
- 2019
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Ingår i: Indiana University Mathematics Journal. - : INDIANA UNIV MATH JOURNAL. - 0022-2518 .- 1943-5258. ; 68:4, s. 1113-1148
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Tidskriftsartikel (refereegranskat)abstract
- We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, div(A(s) (x, u)del u) = div (f) over bar (x), where A(s) (x, u) has "broken" derivatives of orders s >= 0, such as A(s)(x, u) = a(x) + b(x) (u(+))(s), with (u(+))(0) being understood as the characteristic function on {u > 0}. The vector (f) over bar (x) is assumed to be C-alpha in case s = 0, and C-1,C-alpha (or higher) in case s > 0. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More precisely, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.
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