Lecture 1 - Introduction to the problem of regularity Giuseppe Di - PowerPoint PPT Presentation

Lecture 2 G OOD NEWS Morrey spaces imply regularity 1 If the RHS has a sign they are necessary for regularity. 2 Morrey spaces are necessary and sufficient for regularity Giuseppe Di Fazio (University of Catania) 30 / 100

Lecture 2 N ECESSITY OF V ERY WEAK SOLUTION In general L p ,λ is not contained in W − 1 , 2 1 This means that - in general - weak solutions do not exist 2 Giuseppe Di Fazio (University of Catania) 31 / 100

Lecture 2 V ERY W EAK S OLUTION This force us to introduce the concept of very weak solution Giuseppe Di Fazio (University of Catania) 32 / 100

Lecture 2 V ERY WEAK SOLUTION D EFINITION (V ERY WEAK SOLUTION ) Let Ω be a bounded domain in R n and let µ be a bounded variation measure in Ω . A function u ∈ L 1 (Ω) is a very weak solution of the Dirichlet problem � Lu = µ in Ω u = 0 on ∂ Ω if, for any ϕ ∈ W 1 , 2 (Ω) ∩ C 0 (Ω) such that L ϕ ∈ C 0 (Ω) , we have 0 ˆ ˆ uL ϕ dx = ϕ d µ. Ω Ω Giuseppe Di Fazio (University of Catania) 33 / 100

Lecture 2 V ERY WEAK SOLUTION R EMARK The class of test functions is non empty by De Giorgi regularity Theorem. Indeed, let ψ be a C 0 (Ω) given function. Then, there exists a h¨ older continuous weak solution ϕ of the equation L ϕ = ψ so ϕ ∈ W 1 , 2 (Ω) 0 and then ϕ ∈ C 0 (Ω) ∩ W 1 , 2 (Ω) . 0 Giuseppe Di Fazio (University of Catania) 34 / 100

Lecture 2 V ERY WEAK SOLUTION R EMARK Very weak solution is unique. Indeed, the homogeneous problem � Lu = 0 in Ω u = 0 on ∂ Ω and show that u = 0 . Let ψ ∈ C 0 (Ω) and ϕ in W 1 , 2 (Ω) ∩ C 0 (Ω) such 0 ˆ ˆ that L ϕ = ψ . Since u L ϕ dx = 0 i.e. u ψ dx = 0 for any Ω Ω continuous ψ , we have u = 0 in Ω . Giuseppe Di Fazio (University of Catania) 35 / 100

Lecture 2 G REEN FUNCTION Very important for linear differential operators is the Green function. Giuseppe Di Fazio (University of Catania) 36 / 100

Lecture 2 G REEN FUNCTION Let us consider the problem � Lu = T in Ω u = 0 on ∂ Ω By the definition of very weak solution, there exists a linear application G : W − 1 , 2 (Ω) → W 1 , 2 (Ω) 0 defined by G ( T ) = u This is what we call the Green operator. Giuseppe Di Fazio (University of Catania) 37 / 100

Lecture 2 G REEN FUNCTION Now, by the local boundedness and the local h¨ older continuity Theorems we have G : W − 1 , 2 (Ω) → W 1 , 2 (Ω) 0 such that for any T ∈ W − 1 , 2 (Ω) the function u = G ( T ) is the unique weak solution in W 1 , 2 (Ω) of the Dirichlet problem. 0 Giuseppe Di Fazio (University of Catania) 38 / 100

Lecture 2 G REEN FUNCTION T HEOREM For any bounded variation measure µ on Ω there exists a unique solution of the equation Lu = µ that is zero on the boundary ∂ Ω . Moreover it belongs to W 1 , p ′ (Ω) for any p > n. 0 Giuseppe Di Fazio (University of Catania) 39 / 100

Lecture 2 G REEN FUNCTION The operator G maps continuously W − 1 , p (Ω) in C 0 (Ω) Indeed, if p > n , by De Giorgi - Nash - Moser Theorem we have G : W − 1 , p (Ω) → C 0 (Ω) and there exists c such that ∀ ψ ∈ W − 1 , p (Ω) max Ω | G ( ψ ) | ≤ c � ψ � − 1 , p ¯ Giuseppe Di Fazio (University of Catania) 40 / 100

Lecture 2 G REEN FUNCTION Then, u is a very weak solution of Lu = µ vanishing on ∂ Ω if and only if ˆ ˆ u ψ dx = G ( ψ ) d µ Ω Ω for all ϕ ∈ C 0 (Ω) and � � � � ˆ ˆ ˆ � � � � u ψ dx � = G ( ψ ) d µ � ≤ c | d µ |� ψ � − 1 , p � � � � � � Ω Ω Ω for all ϕ ∈ C 0 (Ω) . Giuseppe Di Fazio (University of Catania) 41 / 100

Lecture 2 G REEN FUNCTION By density we have ˆ � u � W 1 , p ′ ≤ c | d µ | Ω for p > n . The application µ �→ u is the adjoint of G , i.e. u = G ∗ ( µ ) . Giuseppe Di Fazio (University of Catania) 42 / 100

Lecture 2 G REEN FUNCTION Since G : W − 1 , p → C 0 (Ω) is continuous by duality we have that G ∗ is also continuous from the space M of the measures with bounded variation in Ω to W 1 , p ′ (Ω) . For 0 any µ ∈ M we have G ∗ ( µ ) ∈ W 1 , p ′ (Ω) . 0 Giuseppe Di Fazio (University of Catania) 43 / 100

Lecture 2 W EAK AND VERY WEAK Now we are ready to compare the notions of weak and very weak solutions. T HEOREM (L ITTMAN - S TAMPACCHIA - W EINBERGER ) Let Ω ⊂ R n be a bounded domain and µ be a bounded variation measure. Let u ∈ L 1 (Ω) be the very weak solution of the Dirichlet problem � Lu = µ in Ω u = 0 on ∂ Ω . Then u is a weak solution if and only if µ ∈ W − 1 , 2 (Ω) . Giuseppe Di Fazio (University of Catania) 44 / 100

Lecture 2 W EAK AND VERY WEAK P ROOF . Let u be the unique weak solution in W 1 , 2 (Ω) of equation Lu = µ . We 0 have ˆ ˆ a ij u x i φ x j dx = φ d µ Ω Ω for any φ ∈ W 1 , 2 (Ω) and then, 0 � � ˆ � � φ d µ � ≤ ν �∇ u � 2 �∇ φ � 2 � � � Ω for all φ ∈ W 1 , 2 (Ω) , which means that µ ∈ W − 1 , 2 (Ω) . 0 Giuseppe Di Fazio (University of Catania) 45 / 100

Lecture 2 W EAK AND VERY WEAK P ROOF . Now, if µ ∈ W − 1 , 2 (Ω) there exists f such that µ = div f and then the equation Lu = div f has a weak solution by classical Hilbert space approach. Giuseppe Di Fazio (University of Catania) 46 / 100

Lecture 2 G REEN FUNCTION D EFINITION (G REEN FUNCTION ) If y ∈ Ω the Dirac mass at y , δ y is a bounded variation measure. Then we may consider the very weak solution g ( · , y ) of the Dirichlet problem in the case µ ≡ δ y . Such a function will be called the Green function for the operator L with respect to the domain Ω with pole at y . Giuseppe Di Fazio (University of Catania) 47 / 100

Lecture 2 G REEN FUNCTION R EMARK The Green function satisfies ˆ ˆ g ( x , y ) L ϕ ( x ) dx = ϕ ( x ) d δ y ( x ) Ω Ω for all ϕ ∈ C 0 (Ω) ∩ W 1 , 2 (Ω) such that L ϕ ∈ C 0 (Ω) . Then, by definition 0 of δ y , we have ˆ ϕ ( y ) = g ( x , y ) L ϕ ( x ) dx . Ω Giuseppe Di Fazio (University of Catania) 48 / 100

Lecture 2 G REEN FUNCTION Using the Green function we may represent the weak solution of the Dirichlet problem. Indeed, let ψ ∈ C 0 (Ω) and ϕ be the weak solution of � L ϕ = ψ in Ω ϕ = 0 on ∂ Ω . Since C 0 (Ω) ⊂ W − 1 , 2 (Ω) we have ϕ ∈ W 1 , 2 (Ω) and, by De Giorgi 0 Theorem, ϕ ∈ C 0 (Ω) . This implies ˆ ˆ g ( x , y ) ψ ( x ) dx = ϕ ( x ) d δ y ( x ) Ω Ω and then ˆ ϕ ( y ) = g ( x , y ) ψ ( x ) dx . Ω Giuseppe Di Fazio (University of Catania) 49 / 100

Lecture 2 G REEN FUNCTION PROPERTIES T HEOREM (G R ¨ UTER & W IDMAN ) There exists a unique function G : Ω × Ω → R ∪ {∞} such that 1. G ( x , y ) ≥ 0 where it is defined. 2. For any y ∈ Ω and any r > 0 such that B r ( y ) ⊂ Ω the function G ( · , y ) belongs to W 1 , 2 (Ω \ B r ( y )) ∩ W 1 , 1 (Ω) . 0 3. The following relation holds true ˆ a ij ( x ) G x i ( x , y ) ϕ x j ( x ) dx = ϕ ( y ) Ω for all ϕ ∈ C ∞ 0 (Ω) . Giuseppe Di Fazio (University of Catania) 50 / 100

Lecture 2 P ROPERTIES OF G REEN FUNCTION T HEOREM Moreover, if we set G ( x ) ≡ G ( x , y ) , the function G satisfies the following properties 4. G belongs to the space L n / ( n − 2 ) (Ω) with bounds depending on the w ellipticity and dimension only. 5. ∇ G belongs to the space L n / ( n − 1 ) (Ω) with bounds depending on w the ellipticity and dimension only. n 6. G belongs to the space W 1 , s 0 (Ω) for any 1 ≤ s < n − 1 with bounds depending on the ellipticity, dimension and the exponent s only. Giuseppe Di Fazio (University of Catania) 51 / 100

Lecture 2 P ROPERTIES OF G REEN FUNCTION T HEOREM 7. There exists a positive constant c depending on the ellipticity and dimension only such that G ( x , y ) ≤ c | x − y | 2 − n for all x , y ∈ Ω , x � = y. 8. There exists a positive constant c depending on the ellipticity and dimension only such that G ( x , y ) ≥ c | x − y | 2 − n for all x , y ∈ Ω such that 0 < | x − y | ≤ 1 2 d ( y , ∂ Ω) , x � = y. Giuseppe Di Fazio (University of Catania) 52 / 100

Lecture 2 O NLY ONE G REEN FUNCTION T HEOREM The Green function g defined by Stampacchia and the other one G defined by Gr¨ uter & Widman are the same function. Giuseppe Di Fazio (University of Catania) 53 / 100

Lecture 2 R EPRESENTATION FORMULA We can represent the very weak solution of the Dirichlet problem Giuseppe Di Fazio (University of Catania) 54 / 100

Lecture 2 R EPRESENTATION FORMULA T HEOREM (R EPRESENTATION FORMULA ) Let µ be a bounded variation measure in a bounded domain Ω ⊂ R n ( n ≥ 3 ) and let u ∈ L 1 (Ω) be the very weak solution of the Dirichlet problem � Lu = µ in Ω u = 0 on ∂ Ω . Then, the following representation formula holds true ˆ u ( x ) = g ( x , y ) d µ ( y ) Ω where g ( x , y ) is the Green’s function for the operator L with respect to Ω with pole at y ∈ Ω . Giuseppe Di Fazio (University of Catania) 55 / 100

Lecture 2 R EPRESENTATION FORMULA P ROOF . We know existence and uniqueness. The proof by direct substitution. Let ϕ ∈ W 1 , 2 (Ω) ∩ C 0 (Ω) be such that L ϕ ∈ C 0 (Ω) . Then 0 � ˆ � ˆ ˆ ϕ ( y ) d µ ( y ) = g ( x , y ) L ϕ ( x ) dx d µ ( y ) Ω Ω Ω � ˆ � ˆ = g ( x , y ) d µ ( y ) L ϕ ( x ) dx Ω Ω ˆ = u ( x ) L ϕ ( x ) dx Ω ˆ ˆ u ( x ) L ϕ ( x ) dx = ϕ ( x ) d µ ( x ) for any and then Ω Ω ϕ ∈ W 1 , 2 (Ω) ∩ C 0 (Ω) such that L ϕ ∈ C 0 (Ω) that is the result. 0 Giuseppe Di Fazio (University of Catania) 56 / 100

Lecture 2 R EGULARITY Representation formula will give us important information about the REGULARITY of the very weak solution. Giuseppe Di Fazio (University of Catania) 57 / 100

Lecture 3 Lecture 3 - Sufficient conditions for regularity Giuseppe Di Fazio (University of Catania) 58 / 100

Lecture 3 B UN B UN I will not go to your boring class! I’m on holiday! Giuseppe Di Fazio (University of Catania) 59 / 100

Lecture 3 R EGULARITY OF VERY WEAK SOLUTION T HEOREM Let 0 < λ < n − 2 , f ∈ L 1 ,λ (Ω) and let u be the very weak solution of the Dirichlet problem � Lu = f in Ω u = 0 on ∂ Ω Then, u ∈ L p λ ,λ (Ω) where w 1 2 = 1 − n − λ . p λ In particular, u ∈ L p (Ω) for any 1 ≤ p < p λ . Moreover, there exists c ≥ 0 such that � u � L p ≤ c � f � L 1 ,λ where c does not depend on u and f. Giuseppe Di Fazio (University of Catania) 60 / 100

Lecture 3 R EGULARITY OF VERY WEAK SOLUTION The proof easily follows from Chiarenza – Frasca Theorem. Giuseppe Di Fazio (University of Catania) 61 / 100

Lecture 3 C HIARENZA – F RASCA T HEOREM T HEOREM (C HIARENZA – F RASCA ) Let 1 < p < + ∞ and 0 < λ < n. Then, there exists a constant c which depend on n, p and λ such that � Mf � L p ,λ ≤ c � f � L p ,λ . If p = 1 we have the following weak type estimate t | { y ∈ B r ( x ) : Mf ( y ) > t } | ≤ c r λ � f � L 1 ,λ . For 1 ≤ p ≤ + ∞ , 0 < λ < n the function Mf is finite for almost all x ∈ R n . Giuseppe Di Fazio (University of Catania) 62 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . By Representation Formula we have ˆ ˆ | x − y | 2 − n | f ( y ) | dy | u ( x ) | ≤ g ( x , y ) | f ( y ) | dy ≤ c Ω Ω a.e. in Ω where g ( x , y ) is the Green’s function of L with respect to Ω with pole at y ∈ Ω and c is a constant which depends on n and ν . Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . Since B r ( x ) ⊂ B 2 r ( x ) ⊂ Ω we have ˆ | f ( y ) | ˆ | f ( y ) | | x − y | n − 2 dy = | x − y | n − 2 dy + Ω B 2 r ( x ) | f ( y ) | ˆ + | x − y | n − 2 dy ≡ I + II . Ω \ B 2 r ( x ) Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . We separately estimate the two integrals by using Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . We have ∞ | f ( y ) | ˆ � I = | x − y | n − 2 dy ≤ B ( x ; r / 2 k − 1 ) \ B ( x ; r / 2 k ) k = 0 � r ∞ � 2 � Mf ( x ) = c r 2 Mf ( x ) ≤ c 2 k k = 0 where Mf ( x ) is the Hardy-Littlewood maximal function of f at x ∈ Ω . Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . Now estimate II . We have ∞ | f ( y ) | ˆ � II = | x − y | n − 2 dy ≤ k = 1 r 2 k + 1 ≤| x − y | < r 2 k ∞ � λ − n + 2 � � 2 k r � f � 1 ,λ = c r λ − n + 2 � f � 1 ,λ . ≤ c k = 0 Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . Then, for any r > 0, ˆ | f ( y ) | | x − y | n − 2 dy ≤ C n ,λ ψ ( r ) Ω where ψ ( r ) ≡ r 2 Mf ( x ) + r λ − n + 2 � f � 1 ,λ − n + 2 for r > 0. Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . By taking the minimum of the right hand side we get | f ( y ) | 2 ˆ | x − y | n − 2 dy ≤ c ( Mf ( x )) 1 / p λ � f � n − λ 1 ,λ Ω Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 P ROOF OF REGULARITY RESULT P ROOF . so that ˆ | f ( y ) | 2 1 ,λ ( Mf ( x )) 1 / p λ n − λ | u ( x ) | ≤ c | x − y | n − 2 dy ≤ c � f � Ω a.e. in Ω and the result follows by Chiarenza - Frasca Theorem. Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3 A COUNTEREXAMPLE If λ → n − 2 then p λ → ∞ . Unfortunately the implication f ∈ L 1 , n − 2 (Ω) ⇒ u ∈ L ∞ (Ω) is not true! Let Ω = { x ∈ R n : 0 < | x | < 1 } , n ≥ 3. Let us check that the very weak solution of the Dirichlet problem ∆ u = n − 2  in Ω  | x | 2 u = 0 on ∂ Ω  is the function u ( x ) = log | x | . Giuseppe Di Fazio (University of Catania) 64 / 100

Lecture 3 V ERY WEAK SOLUTION IN BMO T HEOREM If f ∈ L 1 , n − 2 (Ω) the very weak solution u belongs to BMO locally in the following sense. Let Ω ′ ⋐ Ω and d = dist (Ω ′ , ∂ Ω) . Then, there exists a constant C ≡ C ( n , ν, d ) > 0 such that � dy ≤ C � f � 1 , n − 2 . � � � u ( y ) − u B r ( x ) B r ( x ) for all 0 < r < d 2 and x ∈ Ω ′ . Giuseppe Di Fazio (University of Catania) 65 / 100

Lecture 3 V ERY WEAK SOLUTION = WEAK SOLUTION T HEOREM ˜ S (Ω) ⊂ W − 1 , 2 (Ω) P ROOF . Let f ∈ ˜ S (Ω) and ϕ ∈ C ∞ 0 (Ω) . We have to show that there exists a positive constant C such that |� f , ϕ �| ≤ C φ ( f ) �∇ ϕ � 2 | f ( x ) | Set I 1 ( f )( y ) = ´ | x − y | n − 1 dx . Then Ω � ˆ |∇ ϕ ( y ) | � ˆ |� f , ϕ �| ≤ C | f ( x ) | | x − y | n − 1 dy dx Ω Ω ˆ = C |∇ ϕ ( y ) | I 1 ( f )( y ) dy Ω Giuseppe Di Fazio (University of Catania) 66 / 100 ≤ C �∇ ϕ � L 2 � I 1 ( f ) � L 2 .

Lecture 3 V ERY WEAK SOLUTION = WEAK SOLUTION T HEOREM ˜ S (Ω) ⊂ W − 1 , 2 (Ω) P ROOF . But � ˆ | f ( x ) | � � ˆ | f ( z ) | � ˆ � I 1 ( f ) � 2 = | x − y | n − 1 dx | z − y | n − 1 dz dy Ω Ω Ω � ˆ � ˆ � � ˆ dy = | f ( z ) | | f ( x ) | dx dz . | x − y | n − 1 | z − y | n − 1 Ω Ω Ω Giuseppe Di Fazio (University of Catania) 66 / 100

Lecture 3 V ERY WEAK SOLUTION = WEAK SOLUTION T HEOREM ˜ S (Ω) ⊂ W − 1 , 2 (Ω) P ROOF . By well known properties of Riesz potentials we get | f ( x ) | ˆ ˆ � I 1 ( f ) � 2 L 2 (Ω) ≤ C | f ( z ) | | x − z | n − 2 dx dz . Ω Ω Since f ∈ ˜ S (Ω) , we have ˆ � I 1 ( f ) � 2 L 2 (Ω) ≤ C | f ( x ) | dx < ∞ Ω Thus, the conclusion follows putting together the previous inequalities. Giuseppe Di Fazio (University of Catania) 66 / 100

Lecture 3 V ERY WEAK SOLUTION = B OUNDED WEAK SOLUTION T HEOREM If f ∈ ˜ S (Ω) then the ( weak ) solution u is bounded in Ω . P ROOF . For any x ∈ Ω we have ˆ ˆ | f ( y ) || x − y | 2 − n dy | u ( x ) | ≤ g ( x , y ) | f ( y ) | dy ≤ c Ω Ω ˆ | f ( y ) || x − y | 2 − n dy ≤ c sup r > 0 Ω ∩ B r ( x ) ˆ | f ( y ) || x − y | 2 − n dy . ≤ c sup r > 0 x ∈ Ω Ω ∩ B r ( x ) Giuseppe Di Fazio (University of Catania) 67 / 100

Lecture 3 W EAK SOLUTION IS CONTINUOUS T HEOREM (C HIARENZA – F ABES – G AROFALO ) If f ∈ S (Ω) , then any weak solution u of equation Lu = f is continuous in Ω . Giuseppe Di Fazio (University of Catania) 68 / 100

Lecture 3 U SEFUL INEQUALIES T HEOREM (C ACCIOPPOLI ) Let u be a weak solution of equation Lu = 0 . Then there exists a constant c = c ( n , ν ) such that ˆ ˆ |∇ u | 2 ϕ 2 dx ≤ u 2 |∇ ϕ | 2 dx ∀ ϕ ∈ D (Ω) . Ω Ω T HEOREM (H ARNACK ) Let u be a non negative weak solution of equation Lu = 0 . Then there exists a constant c = c ( n , ν ) such that, for any ball B such that 2 B ⋐ Ω we have sup u ≤ c inf B u B Giuseppe Di Fazio (University of Catania) 69 / 100

Lecture 3 P ROOF . Let η be the Stummel modulus of f . By the embedding the solution is weak and bounded. We have ˆ ˆ A ( x ) ∇ u ∇ ψ dx = f ( x ) ψ ( x ) dx Ω Ω for all ψ ∈ C ∞ 0 (Ω) . If B r is a ball such that B 4 r ⋐ Ω let φ be a cut-off function C ∞ 0 (Ω) such that 0 ≤ φ ≤ 1 in Ω , φ ≡ 1 in B 3 r / 2 , φ ≡ 0 out of B 2 r . Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Then u φ is a weak solutions of L ( u φ ) = f φ − div ( A ( x ) u ∇ φ ) − A ( x ) ∇ u ∇ φ . Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . By representation formula we get ˆ u ( x ) φ ( x ) = f ( y ) φ ( y ) g ( x , y ) dy + Ω ˆ + ∇ y g ( x , y ) A ( y ) u ( y ) ∇ φ ( y ) dy Ω ˆ − ∇ u ( y ) A ( y ) ∇ φ ( y ) g ( x , y ) dy . Ω Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . For any x ∈ B r / 2 ( x 0 ) ˆ u ( x ) − u ( x 0 ) = f ( y ) φ ( y ) ( g ( x , y ) − g ( x 0 , y )) dy Ω ˆ − A ( y ) ∇ u ∇ φ ( g ( x , y ) − g ( x 0 , y )) dy Ω ˆ � � + ∇ g y ( x , y ) − ∇ g y ( x 0 , y ) A ( y ) u ( y ) ∇ φ dy Ω ≡ I + II + III Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . First estimate I . Let N > 1 to be chosen later. ˆ | I | ≤ | f ( y ) φ ( y ) ( g ( x , y ) − g ( x 0 , y )) | dy { y ∈ Ω: | x 0 − y | > N | x − x 0 |} ˆ + | f ( y ) φ ( y ) ( g ( x , y ) − g ( x 0 , y )) | dy { y ∈ Ω: | x 0 − y |≤ N | x − x 0 |} ≡ A + B . To estimate A we use the fact that the Green’s function g ( · , y ) is α -H¨ older continuous out of the pole because of the De Giorgi Theorem Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Namely, the following inequality holds true | g ( x , y ) − g ( x 0 , y ) | � 1 � α � � | x 0 − x | 2 | g ( x , y ) | 2 dx ≤ C r B r ≤ CN − α max x ∈ B r g ( x , y ) ≤ CN − α min x ∈ B r g ( x , y ) ≤ CN − α g ( x 0 , y ) ≤ CN − α | x 0 − y | 2 − n Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . and then ˆ A ≤ CN − α | f ( y ) | φ ( y ) | x 0 − y | 2 − n dy ≤ CN − α η ( 2 r ) . B 2 r ( x 0 ) Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Now estimate B by using Gr¨ uter & Widman Theorem. � � 1 1 | g ( x , y ) − g ( x 0 , y ) | ≤ C | x − y | n − 2 + | x 0 − y | n − 2 Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Then, due to the domain of integration, | f ( y ) | | f ( y ) | ˆ ˆ B ≤ C | x − y | n − 2 dy + | x 0 − y | n − 2 dy | x 0 − y |≤ N | x − x 0 | | x 0 − y |≤ N | x − x 0 | | f ( y ) | ˆ ≤ C | x − y | n − 2 dy + η ( N | x − x 0 | ) ≤ | x − y |≤ ( N + 1 ) | x − x 0 | ≤ C η (( N + 1 ) | x 0 − x | ) + η ( N | x − x 0 | ) Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . � 1 � r 2 By choosing now N = we get | x − x 0 | � α/ 2 � | x − x 0 | | I | ≤ η ( 2 r )+ r � � + η ( r | x − x 0 | ) + η ( r | x − x 0 | + | x − x 0 | ) . Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Now we estimate II and III . ˆ II = ( g ( x , y ) − g ( x 0 , y )) A ( y ) ∇ u ∇ ϕ dy B 2 r \ B 3 r / 2 By De Giorgi Theorem there exists α ≡ α ( n , ν ) > 0 such that � α � | x − x 0 | 1 | g ( x , y ) − g ( x 0 , y ) | ≤ c | x 0 − y | n − 2 r if y ∈ B 2 r \ B 3 r / 2 , so that Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . � α ˆ � | x − x 0 | | II | ≤ c |∇ u | | x 0 − y | n − 2 dy r r B 2 r \ B 3 r / 2 � α ˆ � | x − x 0 | ≤ cr 1 − n |∇ u | dy r B 2 r and then � 1 � α � | x − x 0 | � 2 |∇ u | 2 dy | II | ≤ c ( n , ν ) r . r B 2 r Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Then, by Caccioppoli inequality � 1 � α � � | x − x 0 | 2 | u | 2 dy | II | ≤ c ( n , ν ) . r B 4 r Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Finally we estimate III . ˆ � � III = ∇ g y ( x , y ) − ∇ g y ( x 0 , y ) A ( y ) ∇ ϕ u ( y ) dy B 2 r \ B 3 r / 2 Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . By Cauchy Schwarz inequality and Caccioppoli inequality we have | III | ≤ c ˆ |∇ g y ( x , y ) − ∇ g y ( x 0 , y ) || u | dy r B 2 r \ B 3 r / 2 � 1 � 1 � ˆ 2 � ˆ ≤ c 2 | u | 2 dy |∇ g y ( x , y ) − ∇ g y ( x 0 , y ) | 2 dy r B 2 r B 2 r \ B 3 r / 2 � 1 � 1 � 2 = c � ˆ ˆ 2 | u | 2 dy | g ( x , y ) − g ( x 0 , y ) | 2 dy r 2 4 r < | x 0 − y | < 9 3 B 2 r 4 r Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . De Giorgi Theorem and pointwise estimates of Green function yield � 1 � α � � | x − x 0 | 2 | u | 2 dy | III | ≤ c r B 2 r Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 P ROOF . Merging previous estimates we get � � α/ 2 � | x − x 0 | � | u ( x ) − u ( x 0 ) | ≤ c η ( 2 r ) + η ( r | x − x 0 | ) r � + η ( r | x − x 0 | + | x − x 0 | ) � 1 � α � � � | x − x 0 | 2 | u | 2 dy + → 0 r B 2 r as x → x 0 because f ∈ S (Ω) . Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3 W EAK SOLUTION IS H ¨ OLDER CONTINUOUS T HEOREM If f ∈ L 1 ,λ (Ω) , n − 2 < λ < n, then any weak solution u of Lu = f belong to C 0 ,α (Ω) where α ≡ α ( n , λ, ν, � f � 1 ,λ ) . Giuseppe Di Fazio (University of Catania) 71 / 100

Lecture 3 P ROOF . It is a refinement of the previous result because L 1 ,λ is contained in S . In order to show the result we use the fact that the function f belongs to the Morrey space L 1 ,λ with n − 2 < λ < n that implies the following estimate η ( r ) ≤ c � f � 1 ,λ r λ − n + 2 . Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 3 P ROOF . By using the estimate we finally get �� | x − x 0 | � α/ 2 r λ − n + 2 + | u ( x ) − u ( x 0 ) | ≤ c � f � 1 ,λ r � λ − n + 2 � �� + r | x − x 0 | + | x − x 0 | � 1 � α � � | x − x 0 | 2 | u | 2 dy + c r B 2 r Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 3 P ROOF . and then | u ( x ) − u ( x 0 ) | ≤ c | x − x 0 | β where β = 1 2 min ( λ − n + 2 , α ) where α is the H¨ older exponent of the elliptic operator L arising from De Giorgi Theorem. Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 4 N ECESSARY CONDITIONS FOR REGULARITY Lecture 4 - Necessary conditions for regularity Giuseppe Di Fazio (University of Catania) 73 / 100