| 1. |
- Son, Ora, et al.
(författare)
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ATHB12, an ABA-Inducible Homeodomain-Leucine Zipper (HD-Zip) Protein of Arabidopsis, Negatively Regulates the Growth of the Inflorescence Stem by Decreasing the Expression of a Gibberellin 20-Oxidase Gene
- 2010
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Ingår i: Plant and Cell Physiology. - : Oxford University Press (OUP). - 0032-0781 .- 1471-9053. ; 51:9, s. 1537-1547
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Tidskriftsartikel (refereegranskat)abstract
- Arabidopsis thaliana homeobox 12 (ATHB12) is rapidly induced by ABA and water stress. A T-DNA insertion mutant of ATHB12 with a reduced level of ATHB12 expression in stems had longer inflorescence stems and reduced sensitivity to ABA during germination. A high level of transcripts of gibberellin 20-oxidase 1 (GA20ox1), a key enzyme in the synthesis of gibberellins, was detected in athb12 stems, while transgenic lines overexpressing ATHB12 (A12OX) had a reduced level of GA20ox1 in stems. Consistent with these data, ABA treatment of wild-type plants resulted in decreased GA20ox1 expression whereas ABA treatment of the athb12 mutant gave rise to slightly decreased GA20ox1 expression. Retarded stem growth in 3-week-old A12OX plants was rescued by exogenous GA(9), but not by GA(12), and less GA(9) was detected in A12OX stems than in wild-type stems. These data imply that ATHB12 decreases GA20ox1 expression in stems. On the other hand, the stems of A12OX plants grew rapidly after the first 3 weeks, so that they were almost as high as wild-type plants at about 5 weeks after germination. We also found changes in the stems of transgenic plants overexpressing ATHB12, such as alterations of expression GA20ox and GA3ox genes, and of GA(4) levels, which appear to result from feedback regulation. Repression of GA20ox1 by ATHB12 was confirmed by transfection of leaf protoplasts. ABA-treated protoplasts also showed increased ATHB12 expression and reduced GA20ox1 expression. These findings all suggest that ATHB12 negatively regulates the expression of a GA 20-oxidase gene in inflorescence stems.
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| 2. |
- 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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| 3. |
- Figalli, Alessio, et al.
(författare)
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Constraint maps with free boundaries : the obstacle case
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As already pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, is the properties of the projected image and its higher regularity, which we show to be at most of class C2,1. In arbitrary dimensions, we prove that the image map is globally of class W3,BMO, and locally of class C2,1 around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.
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| 4. |
- Figalli, Alessio, et al.
(författare)
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Constraint maps with free boundaries : the Bernoulli case
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this manuscript, we delve into the study of maps that minimize the Alt–Caffarelli energy functional∫Ω(|Du|2 + q2χu−1(M )) dx,under the condition that the image u(Ω) is confined within ¯M . Here, Ω denotes a bounded domain in the ambient space Rn (with n ≥ 1), and M represents a smooth domain in the target space Rm (where m ≥ 2).Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, int(u−1(∂M )), such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary.Our first significant contribution is the validity of a ε-regularity theorem. This theorem is founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small energy.Our subsequent key finding reveals that, whenever the complement of M is uniformly convexand of class C3, the maps minimizing the Alt–Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the non-coincidence set u−1(M ). In particular, this Lipschitz continuity extends to the free boundary.A noteworthy consequence of our findings is the smoothness of flat free boundaries and of theresulting image maps.
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| 5. |
- 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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| 6. |
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| 7. |
- 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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| 8. |
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| 9. |
- 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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| 10. |
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| 11. |
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| 12. |
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| 13. |
- 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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| 14. |
- 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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| 15. |
- 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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| 16. |
- 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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| 17. |
- 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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| 18. |
- 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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| 19. |
- Kim, Sunghan
(författare)
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Uniform Integrability in Periodic Homogenization of Fully Nonlinear Elliptic Equations
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This paper is devoted to the study of uniform W1,np/n-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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| 20. |
- 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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