� Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients Norayr Matevosyan Arshak Petrosyan Kolmogorov Equations in Physics and Finance September 9, 2010 Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 1 / 26
Original Elliptic Monotonicity Formula Teorem (Alt-Caffarelli-Friedman 1984) Let u ± be two continuous functions in B in R n such that u ± ≥ , ∆ u ± ≥ , u + ⋅ u − = in B u + > ∆ u + ≥ then the functional ∣∇ u + ∣ ∣∇ u − ∣ u − > φ ( r ) = φ ( r , u + , u − ) = ∣ x ∣ n − dx ∫ B r r ∫ B r ∣ x ∣ n − dx ∆ u − ≥ is monotone nondecreasing in r ∈ ( , ] . Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 2 / 26
Original Elliptic Monotonicity Formula Teorem (Alt-Caffarelli-Friedman 1984) Let u ± be two continuous functions in B in R n such that u + ⋅ u − = u ± ≥ , ∆ u ± ≥ , in B u + > ∆ u + ≥ then the functional ∣∇ u + ∣ ∣∇ u − ∣ u − > φ ( r ) = φ ( r , u + , u − ) = ∣ x ∣ n − dx ∫ B r r ∫ B r ∣ x ∣ n − dx ∆ u − ≥ is monotone nondecreasing in r ∈ ( , ] . Tis formula has been of fundamental importance in the regularity theory of free boundaries, especially in problems with two phases. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 2 / 26
Original Elliptic Monotonicity Formula One of the applications of the monotonicity formula is the ability to produce estimates of the type c n ∣ ∇ u + ( )∣ ∣ ∇ u − ( )∣ ≤ φ ( + ) ≤ φ ( / ) ≤ C n ∥ u + ∥ L ( B ) ∥ u − ∥ L ( B ) Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 3 / 26
Original Elliptic Monotonicity Formula One of the applications of the monotonicity formula is the ability to produce estimates of the type c n ∣ ∇ u + ( )∣ ∣ ∇ u − ( )∣ ≤ φ ( + ) ≤ φ ( / ) ≤ C n ∥ u + ∥ L ( B ) ∥ u − ∥ L ( B ) Te proof is based on the following eigenvalue inequality of Friedland-Hayman 1976. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 3 / 26
Original Elliptic Monotonicity Formula One of the applications of the monotonicity formula is the ability to produce estimates of the type c n ∣ ∇ u + ( )∣ ∣ ∇ u − ( )∣ ≤ φ ( + ) ≤ φ ( / ) ≤ C n ∥ u + ∥ L ( B ) ∥ u − ∥ L ( B ) Te proof is based on the following eigenvalue inequality of Friedland-Hayman 1976. For Σ ⊂ ∂B define λ ( Σ ) = inf ∫ Σ ∣ ∇ θ f ∣ f ∣ ∂ Σ = , ∫ Σ f Define also α ( Σ ) so that λ ( Σ ) = α ( Σ )( n − + α ( Σ )) . Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 3 / 26
Original Elliptic Monotonicity Formula One of the applications of the monotonicity formula is the ability to produce estimates of the type c n ∣ ∇ u + ( )∣ ∣ ∇ u − ( )∣ ≤ φ ( + ) ≤ φ ( / ) ≤ C n ∥ u + ∥ L ( B ) ∥ u − ∥ L ( B ) Te proof is based on the following eigenvalue inequality of Friedland-Hayman 1976. For Σ ⊂ ∂B define λ ( Σ ) = inf ∫ Σ ∣ ∇ θ f ∣ f ∣ ∂ Σ = , ∫ Σ f Define also α ( Σ ) so that λ ( Σ ) = α ( Σ )( n − + α ( Σ )) . Teorem (Friedland-Hayman 1976) Let Σ ± be disjoint open sets on ∂B . Ten α ( Σ + ) + α ( Σ − ) ≥ . Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 3 / 26
Parabolic Monotonicity Formula Teorem (Caffarelli 1993) Let u ± ( x , s ) be two continuous functions in S = R n × (− , ] u + ⋅ u − = u ± ≥ , ( ∆ − ∂ s ) u ± ≥ , in S then − r ∫ R n ∣∇ u + ∣ G ( x , − s ) dxds ∫ − r ∫ R n ∣∇ u − ∣ G ( x , − s ) dxds Φ ( r , u + , u − ) = r ∫ is monotone nondecreasing for r ∈ ( , ] . Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 4 / 26
Parabolic Monotonicity Formula Teorem (Caffarelli 1993) Let u ± ( x , s ) be two continuous functions in S = R n × (− , ] u + ⋅ u − = u ± ≥ , ( ∆ − ∂ s ) u ± ≥ , in S then − r ∫ R n ∣∇ u + ∣ G ( x , − s ) dxds ∫ − r ∫ R n ∣∇ u − ∣ G ( x , − s ) dxds Φ ( r , u + , u − ) = r ∫ is monotone nondecreasing for r ∈ ( , ] . Note that u ± must be defined in a entire strip and we must have a moderate growth at infinity. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 4 / 26
Parabolic Monotonicity Formula Te proof is now based on the eigenvalue inequality in Gaussian space. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 5 / 26
Parabolic Monotonicity Formula Te proof is now based on the eigenvalue inequality in Gaussian space. For Ω ⊂ R n define λ ( Ω ) = inf ∫ Ω ∣ ∇ f ∣ dν dν = ( π ) − n / e − x / dx . ∫ Ω f dν , Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 5 / 26
Parabolic Monotonicity Formula Te proof is now based on the eigenvalue inequality in Gaussian space. For Ω ⊂ R n define λ ( Ω ) = inf ∫ Ω ∣ ∇ f ∣ dν dν = ( π ) − n / e − x / dx . ∫ Ω f dν , Teorem (Beckner-Kenig-Pipher) Let Ω ± be two disjoint open sets in R n . Ten λ ( Ω + ) + λ ( Ω − ) ≥ Te proof is reduced to the convexity result of Brascamp-Lieb 1976 for first eigenvalues of − ∆ + V ( x ) with convex potential V as a function of the domain. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 5 / 26
Localized Parabolic Formula Teorem (Caffarelli 1993) Let u ± ( x , s ) be two continuous functions in Q − = B × (− , ] such that u + ⋅ u − = u ± ≥ , ( ∆ − ∂ s ) u ± ≥ , Q − in . Let ψ ∈ C ∞ ( B ) be a cutoff function such that ψ ∣ B / = ≤ ψ ≤ , supp ψ ⊂ B / , then Φ ( r ) = Φ ( r , u + ψ , u − ψ ) is almost monotone in a sense that Φ ( +) − Φ ( r ) ≤ Ce − c / r ∥ u + ∥ ) ∥ u − ∥ ) . L ( Q − L ( Q − Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 6 / 26
Generalization: Caffarelli-Kenig Estimate Instead of the heat operator ∆ − ∂ s consider now uniformly parabolic L u = L A , b , c u ∶= div (A( x , s )∇ u ) + b ( x , s ) ⋅ ∇ u + c ( x , s ) u − ∂ s u Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 7 / 26
Generalization: Caffarelli-Kenig Estimate Instead of the heat operator ∆ − ∂ s consider now uniformly parabolic L u = L A , b , c u ∶= div (A( x , s )∇ u ) + b ( x , s ) ⋅ ∇ u + c ( x , s ) u − ∂ s u Assume A to be Dini continuous, b , c uniformly bounded Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 7 / 26
Generalization: Caffarelli-Kenig Estimate Instead of the heat operator ∆ − ∂ s consider now uniformly parabolic L u = L A , b , c u ∶= div (A( x , s )∇ u ) + b ( x , s ) ⋅ ∇ u + c ( x , s ) u − ∂ s u Assume A to be Dini continuous, b , c uniformly bounded Teorem (Caffarelli-Kenig 1998) Let u ± ( x , s ) be two continuous functions in Q − such that u + ⋅ u − = u ± ≥ , L u ± ≥ , Q − in . Let ψ ∈ C ∞ ( B ) be a cutoff function as before. Ten Φ ( r ) = Φ ( r , u + ψ , u − ψ ) is almost monotone in a sense that we have an estimate ) + ∥ u − ∥ Φ ( r ) ≤ C (∥ u + ∥ ) ) r < r . , L ( Q − L ( Q − Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 7 / 26
Generalization: Caffarelli-Jerison-Kenig Estimate Teorem (Caffarelli-Jerison-Kenig 2002) Let u ± be two continuous functions in B in R n such that u + ⋅ u − = u ± ≥ , ∆ u ± ≥ − , in B then the functional φ ( r ) = φ ( r , u + , u − ) satisfies φ ( r ) ≤ C ( + ∥ u + ∥ L ( B ) + ∥ u − ∥ L ( B ) ) r < r . , Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 8 / 26
Generalization: Caffarelli-Jerison-Kenig Estimate Teorem (Caffarelli-Jerison-Kenig 2002) Let u ± be two continuous functions in B in R n such that u + ⋅ u − = u ± ≥ , ∆ u ± ≥ − , in B then the functional φ ( r ) = φ ( r , u + , u − ) satisfies φ ( r ) ≤ C ( + ∥ u + ∥ L ( B ) + ∥ u − ∥ L ( B ) ) r < r . , Te proof is based on a sophisticated iteration scheme. Matevosyan, Petrosyan (Cambridge, Purdue) Almost monotonicity formulas PIMS 8 / 26
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