| 1. |
- Colombo, Maria, et al.
(författare)
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A transmission problem with (p, q)-Laplacian
- 2023
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 48:2, s. 315-349
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the so-called double-phase problem where the phase transition takes place across the interface of the positive and negative phase of minimizers of the functional (Formula presented.) We prove that minimizers exist, are Hölder regular and verify (Formula presented.) in a weak sense. We also prove that their free boundary is (Formula presented.) a.e. with respect to the measure (Formula presented.) whose support is of σ-finite (Formula presented.) -dimensional Hausdorff measure.
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| 2. |
- Figalli, Alessio, et al.
(författare)
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Lipschitz regularity in vectorial linear transmission problems
- 2022
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Ingår i: Nonlinear Analysis. - : Elsevier BV. - 0362-546X .- 1873-5215. ; 221, s. 112911-
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Tidskriftsartikel (refereegranskat)abstract
- We consider vector-valued solutions to a linear transmission problem, and we provethat Lipschitz-regularity on one phase is transmitted to the next phase. Moreexactly, given a solutionu:B1 subset of Rn -> Rmto the elliptic systemdiv((A+ (B-A)chi D) backward difference u) = 0inB1,whereAandBare Dini continuous, uniformly elliptic matrices, we prove that if backward difference u is an element of L infinity(D)thenuis Lipschitz inB1/2. A similar result is also derived for theparabolic counterpart of this problem.
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| 3. |
- Ghergu, Marius, et al.
(författare)
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Isolated singularities for semilinear elliptic systems with power-law nonlinearity
- 2020
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Ingår i: Analysis & PDE. - : MATHEMATICAL SCIENCE PUBL. - 2157-5045 .- 1948-206X. ; 13:3, s. 701-739
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Tidskriftsartikel (refereegranskat)abstract
- We study the system -Delta u = vertical bar u vertical bar(alpha-1) with 1 < alpha <= n+2/n-2, where u = (u(1),...u(m)), m >= 1, is a C-2 nonnegative function that develops an isolated singularity in a domain of R-n, n >= 3. Due to the multiplicity of the components of u, we observe a new Pohozaev invariant different than the usual one in the scalar case. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On a technical level, we adopt the method of moving spheres and the balanced-energy-type monotonicity functionals.
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| 4. |
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| 5. |
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| 6. |
- Kim, Sunghan, et al.
(författare)
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Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems
- 2024
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Ingår i: Journal des Mathématiques Pures et Appliquées. - : Elsevier. - 0021-7824 .- 1776-3371. ; 185, s. 1-46
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Tidskriftsartikel (refereegranskat)abstract
- We prove new optimal C1,α regularity results for obstacle problems involving evolutionary p-Laplace type operators in the degenerate regime p > 2. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward p-paraboloids, up to the critical exponent, α ≤ 2/(p − 2), and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or C1,α intrinsic interpolative geometries.
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| 7. |
- Kim, Sunghan
(författare)
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Nodal sets for broken quasilinear partial differential equations with Dini coefficients
- 2021
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 46:10, s. 1973-2014
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Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, (Formula presented.) where (Formula presented.) and (Formula presented.) are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that (Formula presented.) and (Formula presented.) are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below Hölder regime, and as a result, we establish a structure theorem for singular sets. We also estimate the Hausdorff measure of nodal sets, provided that the vanishing order of given solution is bounded throughout its nodal set, via an approach that extends the classical argument to certain solutions with discontinuous gradient. Besides, we also prove Lipschitz regularity of solutions and continuous differentiability of their nodal set around regular points.
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| 8. |
- Kim, Sunghan, et al.
(författare)
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Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations
- 2022
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Nature. - 0003-9527 .- 1432-0673. ; 243:2, s. 697-745
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Tidskriftsartikel (refereegranskat)abstract
- This article is concerned with uniform C1,α and C1 , 1 estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.
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| 9. |
- Kim, Sunghan
(författare)
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Uniform integrability in periodic homogenization of fully nonlinear elliptic equations
- 2023
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Ingår i: Annali di Matematica Pura ed Applicata. - : Springer Nature. - 0373-3114 .- 1618-1891. ; 202:6, s. 2585-2627
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Tidskriftsartikel (refereegranskat)abstract
- This paper is devoted to the study of uniform W1,npn-p- and W2,p-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.
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