| 1. |
- Andersson, John, et al.
(författare)
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On the singularities of a free boundary through Fourier expansion
- 2012
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 187:3, s. 535-587
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.
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| 2. |
- Andersson, John
(författare)
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Optimal regularity for the Signorini problem and its free boundary
- 2016
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Ingår i: Inventiones Mathematicae. - : Springer Nature. - 0020-9910 .- 1432-1297. ; 204:1, s. 1-82
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Tidskriftsartikel (refereegranskat)abstract
- We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if (Formula presented.) minimizes (Formula presented.)in the convex set (Formula presented.)where (Formula presented.) say. Then (Formula presented.). Moreover the free boundary, given by (Formula presented.)will be a (Formula presented.) graph close to points where (Formula presented.) is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.
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| 3. |
- Andersson, Mats, 1957, et al.
(författare)
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A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas
- 2012
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 190:2, s. 261-297
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Tidskriftsartikel (refereegranskat)abstract
- Let $X$ be a complex space of pure dimension. We introduce fine sheaves $\A^X_q$ of $(0,q)$-currents, which coincides with the sheaves of smooth forms on the regular part of $X$, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\hol^X$. Our construction is based on intrinsic and quite explicit semi-global Koppelman formulas.
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| 4. |
- Andersson, Mats, 1957, et al.
(författare)
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Global effective versions of the Briancon-Skoda-Huneke theorem
- 2015
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 200:2, s. 607-651
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Tidskriftsartikel (refereegranskat)abstract
- We prove global effective versions of the Briançon–Skoda–Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in Cn, and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.
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| 5. |
- Auscher, Pascal, et al.
(författare)
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Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
- 2011
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Ingår i: Inventiones Mathematicae. - : Springer. - 0020-9910 .- 1432-1297. ; 184:1, s. 47-115
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Tidskriftsartikel (refereegranskat)abstract
- We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$. Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension $3$ or higher. The existence of a proof {\em a priori} to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.
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| 6. |
- Baladi, Viviane, et al.
(författare)
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Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps
- 2015
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 201:3, s. 773-844
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Tidskriftsartikel (refereegranskat)abstract
- We consider families of nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure of , as a function of on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters , and, for each , a set of polynomially recurrent parameters containing as a Lebesgue density point, and constants , , so that, for every -Holder function , In addition, for all , the renormalisation period of satisfies , and there are uniform bounds on the rates of mixing of for all with . If , the set contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter , we find a set of CE parameters accumulating at , a constant , and a function , so that C vertical bar t - t(0)vertical bar(1/2) >= vertical bar integral A(0)d mu(t) - integral A(0)d mu(t0) vertical bar >= C-1 vertical bar t - t(0)vertical bar(1/2), for all t is an element of Delta(MT)'.
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| 7. |
- Bayer-Fluckiger, Eva, et al.
(författare)
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Non unimodular Hermitian forms
- 1996
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 123:2, s. 233-240
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Tidskriftsartikel (refereegranskat)
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| 8. |
- Benedicks, Michael, et al.
(författare)
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Solution of the basin problem for Henon-like attractors
- 2001
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 143:2, s. 375-434
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Tidskriftsartikel (refereegranskat)abstract
- For a large class of non-uniformly hyperbolic attractors of dissipative diffeomorphisms, we prove that there are no holes in the basin of attraction: stable manifolds of points in the attractor fill-in a full Lebesgue measure subset. Then, Lebesgue almost every point in the basin is generic for the SRB (Sinai-Ruelle-Bowen) measure of the attractor. This solves a problem posed by Sinai and by Ruelle, for this class of systems.
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| 9. |
- Berman, Robert, 1976, et al.
(författare)
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Growth of balls of holomorphic sections and energy at equilibrium
- 2010
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 181:2, s. 337-394
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Tidskriftsartikel (refereegranskat)abstract
- Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e (-phi) on L, we define the energy at equilibrium of (K,phi) as the Monge-AmpSre energy of the extremal psh weight associated to (K,phi). We prove the differentiability of the energy at equilibrium with respect to phi, and we show that this energy describes the asymptotic behaviour as k -> a of the volume of the sup-norm unit ball induced by (K,k phi) on the space of global holomorphic sections H (0)(X,kL). As a consequence of these results, we recover and extend Rumely's Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan's equidistribution theorem for algebraic points of small height to the case of a big line bundle.
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| 10. |
- Berman, Robert, 1976
(författare)
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K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
- 2016
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 203:3, s. 973-1025
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einsteinmetrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's lambda-entropy functional on K-unstable Fano manifolds are also given.
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