Regularity of solutions to quasilinear elliptic Historical Notes - PowerPoint PPT Presentation

Summary Elliptic Systems Regularity of solutions to quasilinear elliptic Historical Notes Local systems boudedness Anisotropic behavior p , q -growth Elvira Mascolo Systems with p , q -growth General growth Department of Mathematics “Ulisse Dini” The Last slide University of Firenze Universitá Politecnica delle Marche - Ancona (Italy) Optimization Days, June 6-8, 2011

Elliptic Systems Summary n � Elliptic Systems D x i A α i ( x , u , Du ) = B i ( x , u , Du ) α = 1 , ..., m Historical Notes i = 1 Local boudedness Ω ⊂ R n open bounded, n ≥ 2; Anisotropic behavior p , q -growth A i : Ω × R n × R nm − → R n , B i : Ω × R n × R nm − → R Systems with p , q -growth u ∈ W 1 , 1 (Ω; R m ) weak solution General growth The Last slide m n m � B α ( x , u , Du ) ϕ α dx = 0 � � � A α i ( x , u , Du ) ϕ α x i + Ω α = 1 i = 1 α = 1 for all test function ϕ

Elliptic Systems Summary The definition of weak solution leads to assign growth Elliptic Systems assumptions on A α i and B i Historical Notes Local boudedness Anisotropic Regularity of weak solution behavior p , q -growth Systems with p , q -growth The situation is very different with respect to the General growth single equation case The Last slide There is a gap in the regularity scale for the solutions of systems and for the minimizers of integral vectorial functionals

Historical Notes We confine our presentation to the fundamental steps Summary Elliptic Systems Historical Notes Hadamard 1890, Bernstein 1904, n = m = 2 Local boudedness Anisotropic behavior Contributions of Caccioppoli 1933, Schauder 1934, Morrey 1938, p , q -growth Douglas-Nirenberg 1954 Systems with p , q -growth General growth No real progress was made (except in two dimensional case) until The Last slide De Giorgi 1957 Nash 1958, Parabolic and Elliptic equations: "P . R. Garabedian writes from London of a manuscript by Ennio de Giorgi containing such a result"

Historical Notes Summary Very powerful theory of regularity Elliptic Systems Historical Notes Local Linear elliptic equation and quadratic functionals boudedness Anisotropic behavior p , q -growth a ij ( x ) ∈ L ∞ Systems with p , q -growth General growth a ij ( x ) ξ i ξ j ≥ ν | ξ | 2 The Last slide every weak solution u ∈ W 1 , 2 is locally Hölder continuous

Historical Notes: Single Equation De Giorgi methods are based on different steps Summary Caccioppoli type inequalities on level sets 1 Elliptic Systems Historical Notes Local boundedness 2 Local boudedness Local Hölder continuity 3 Anisotropic behavior p , q -growth Moser 1960 generalizes Harnack inequality to general linear Systems with p , q -growth equations General growth The Last slide Generalizations by: Stampacchia 1958-1960 Ladyzhenskaya and Ural’tseva 1968 papers and book Serrin 1964-1965 complete analysis in nonlinear case and a ∈ W 1 , 2 counterexample to the regularity when u /

Historical Notes: Conterexamples to regularity None of the new proofs given of the De Giorgi’s result could be extended to cover the case of systems Summary Elliptic Systems Historical Notes De Giorgi 1968 proved that his result cannot extended to systems Local boudedness De Giorgi’s counterexample Anisotropic behavior ij ( x ) ∂ u β ∂ � ( a αβ p , q -growth ) = 0 , n = m > 2 ∂ x i ∂ x j Systems with ij p , q -growth General growth | x | 2 ][( n − 2 ) δ j β + x β x j ij ( x ) = δ ij δ αβ + [( n − 2 ) δ i α + x α x i a αβ | x | 2 ] The Last slide | x | γ with γ = n x 1 u ( x ) = 2 ( 1 − ) 4 ( n − 1 ) 2 + 1 � is a solution in R n − { 0 } and a weak solution: u ∈ W 1 , 2 but is not bounded

Historical Notes: Conterexamples to regularity Summary Giusti-Miranda 1968 and Maz´ ja 1968 in the quasilinear case Elliptic Systems Historical Notes For extremals of integral functional (sistems in variation) Local cas 1975 ( n , m = n 2 ) Ne ˇ boudedness Anisotropic Nonlinear case with different growth assumption: Freshe behavior p , q -growth 1973, Hildebrandt-Widman 1975 Systems with More recent contribution by ˇ p , q -growth Sverák-Yan 2000 (n=3, m=5) General growth The Last slide Phenomenon purely vectorial Weak solutions to nonlinear elliptic systems or extremals to vector valued regular integrals in general are not smooth

Regularity for systems Summary These counterexamples suggested two directions in the Elliptic Systems mathematical literature Historical Notes Local boudedness indirect approch to regularity : partial regularity i.e. 1 Anisotropic behavior smootheness of solutions up to a set Ω 0 of zero measure p , q -growth with the study of the properties of the singular set Systems with p , q -growth everywhere regularity in the interior of Ω , when it is General growth 2 The Last slide possible, starting as usual from the local boundedness Bombieri 1976 ....it is an interesting open question to find "good conditions" which imply regularity everywhere

Local boundedness: few contributions Everywhere regularity needs additional assumptions Summary Elliptic Systems Local boudedness of solutions of Linear Elliptic Systems: Historical Notes Local Ladyzhenskaya and Ural’tseva, 1968 boudedness Anisotropic   n n m behavior ∂ ( x ) u β + f α � � � a ij ( x ) u α b αβ  + x j + i ( x ) p , q -growth  i ∂ x i Systems with i = 1 j = 1 β = 1 p , q -growth General growth n m m d αβ ( x ) u β = f α ( x ) � � c αβ ( x ) u β � The Last slide + x i + i i = 1 β = 1 β = 1 , d αβ bounded measurable and ∀ α = 1 , 2 , . . . , m , a ij , b αβ , c αβ i i given functions f α i , f α , Generalizations: Ne ˇ cas-Stara 1972, Tomi 1973, Weigner 1977

Bonn School: Meier results Meier, 1982 in his PhD thesis (supervisor Hildebrandt) and in a subsequent paper studied the boundedness (and integrability Summary Elliptic Systems properties) of solutions to quasilinear elliptic systems: Historical Notes Local div ( A α ( x , u , Du )) = B α ( x , u , Du ) boudedness α = 1 , ... m Anisotropic behavior p , q -growth under the natural conditions: p , p -growth Systems with p , q -growth General growth The Last slide α A α ξ α ≥ | ξ | p − b | u | p − 1 − c 1 � | A α | ≤ C ( | ξ | p − 1 + | u | p − 1 + c 1 ) | B α | ≤ C ( | ξ | p − 1 + | u | p − 1 + c 1 )

Meier Theorem Meier’s result is obtained through the pointwise assumption Summary for the indicator function Elliptic Systems Historical Notes Local Positivity of indicator function boudedness Anisotropic behavior u α u β � | u | 2 Du β A α ( x , u , Du ) ≥ 0 p , q -growth I A ( x , u , Du ) = Systems with α,β p , q -growth General growth The Last slide The arguments of the proof consist in a nontrivial generalization of the Serrin arguments for the single equation The linear case considered by Ladyzhenskaya and Ural’tseva is included

Further contributions Local boundedness for systems Under the same assuptions of Meier additional results by Summary Landes 1989, 2000, 2005 Elliptic Systems Historical Notes Following the ideas of Landes Krömer 2009 obtained similar Local boudedness results to Meier’s ones (which however is not cited) for zero Anisotropic behavior boundary data p , q -growth Systems with p , q -growth The Meier’s condition on I A imposes structure conditions General growth The Last slide Structure conditions | Du | p dx , p ≥ 2 div ( | Du | p − 2 Du ) = 0 � I ( u ) = and Uhlenbeck 1975 gave a complete regularity result: u ∈ C 1 ,τ Giaquinta-Modica 1986, Acerbi-Fusco 1989

Hölder continuity Summary Also for the systems the local boundedness is the first step to get Elliptic Systems more regularity Historical Notes Hölder continuity for BOUNDED solution Local boudedness Anisotropic behavior Under additional structure assumptions: p , q -growth Systems with p , q -growth General growth Wiegner 1976, 1981 The Last slide Hildebrant-Widman 1977: Green’s function Caffarelli 1982 with different methods: weak Harnack inequality for supersolutions of a linear elliptic equation.

Test functions The generalization to systems of the arguments used Summary Elliptic Systems for a single equation are by no means obvious Historical Notes Local boudedness Technical problems depend very often on Anisotropic behavior p , q -growth Systems with the availability of appropriate test functions p , q -growth General growth using the solution as a test function The Last slide the way of truncating the vector valued solution: in the area of truncation the gradient is not vanishing as it does in the scalar case and can interfere in a bad way with the leading part

Quasilinear elliptic system Summary   n n Elliptic Systems ∂  = f α ( x , u , Du ) � � a ij ( x , u , Du ) u α x j + b α i ( x , u , Du ) Historical Notes  ∂ x i Local i = 1 j = 1 boudedness Anisotropic α = 1 , 2 , . . . , m behavior p , q -growth Systems with p , q -growth Generalization of Ladyzhenskaya and Ural’tseva system to General growth the quasilinear case The Last slide Arises in many problems in differential geometry such as that harmonic mappings between manifolds or surfaces of prescribed mean curvature.