Two weighted nonlinear Calderón-Zygmund estimates for nonlinear elliptic equations Tuoc Phan University of Tennessee, Knoxville, TN, USA 33rd Southeastern Analysis Meeting, Knoxville, TN Supported by Simons Foundation: grant # 354889
Outline Introduction - Degenerate nonlinear elliptic equations Weighted Sobolev spaces, and weak solutions Regularity questions, and classical Calderón-Zygmund regularity estimates Asymptotically Uhlenbeck type equations, Sawyer s condition on weights Our main theorem, and ideas in its proof Conclusion, remarks
Let Ω R n be open, bounded, with boundary Ω (could be non-smooth). We study the regularity of weak solutions of the nonlinear elliptic equations { div[a(x, u, u)] = div[f(x)] in Ω, u = g(x) on Ω, u : Ω R n is an unknown solution (in weak sense), F : Ω R n is a given measurable vector field A : Ω R R n R n is a given vector field: (i) A(x,, ) is continuous for a.e. x Ω, A(, z, η) is measurable. (ii) A(, z, η) is could be singular + degenerate: There exists Λ > 0 andµ A 2, Muckenhoupt class of weights, such that A(x, z,η) Λµ(x)η, Λ 1 µ(x) η 2 A(x, z,η),η, for a.e. x Ω, and for all (z,η) R R n. Note: The equation is called uniformly elliptic ifµ=1.
Why degenerate equations are of interest? Some nice analysis and fun to work on Nonlinear equations can be considered as linear degenerate equations. For example: Once we know there is a solution u of the equation div[φ(u) u] = div(f), we can be considered as div[a(x) u] = div(f), when we take a(x) = φ(u(x)). Some non-local PDEs can be studied through some other generate equations. Degenerate equations also appear naturally from some models: Math finance, porous media.
A p -Muckenhoupt weights A non-negative locally integrable functionµ:r n Ris in A p, 1<p< if and only if [µ] Ap := sup ball B R n ( µ(x)dx B )( ) p 1 µ 1 1 p (x)dx <. B In particular,µ A 2 weights if ( [µ] A2 := sup ball B R n µ(x)dx B )( ) µ 1 (x)dx <. B Typical example:µ(x) = x α, thenµ A p if and only if n<α<n(p 1). It turns out that the classa p is monotone in p: A p A q for 1<p< q. Also, observe thatµ A 2, thenµ 1 A 2.
Weighted Sobolev spaces Let us fix p>1, Ω R n, andµ:r n [0, ] is a weight function, andµ A p. Weighted Lebesgue space L p (Ω,µ) consists of all measurable function f : Ω R such that ˆ f p L p (Ω,µ) := f(x) p µ(x)dx<. Ω Weighted Sobolev space W 1,p (Ω,µ) consists of all measurable function f L p (Ω,µ) such that xk f L p (Ω,µ), for k = 1, 2,, n, and f W 1,p (Ω,µ) = f L p (Ω,µ) + n xk f L p (Ω,µ). k=1 W 1,p 0 (Ω,µ) is the closure of C 0 (Ω) in W 1,p (Ω,µ).
Weak solutions Definition Let F such that F/µ L 2 (Ω,µ) and g W 1,2 (Ω,µ) with some fixed µ A 2. A function u W 1,2 Ω,µ) is a weak solution of { div[a(x, u, u)] = div[f], in Ω, u = g, on Ω, if u g W 1,2 (Ω,µ) and 0 ˆ A(x, u, u), ϕ dx = Ω ˆ Ω F, ϕ dx, ϕ W 1,2 0 (Ω,µ).
Regularity question to study For either linear equation div[a(x) u] = div[f], in or more complicated nonlinear one div[a(x, u, u)] = div[f], in Weak solutions: Standard existence theory usually provides solutions in energy space: In our case: u W 1,2 (Ω,µ), given Ω Ω g W 1,2 (Ω,µ), F/µ L 2 (Ω,µ). Regularity question: Given that the data is more regular, for example F/µ L p (Ω,ω), and g W 1,p (Ω,ω) with some other weightωand p 2, will it true that u W 1,p (Ω,ω)? If yes, quantify the estimate (this is known as Calderón-Zygmund estimate)!
Classical CZ estimates for linear uniformly elliptic equations Consider the linear elliptic equation div[a(x) u] = div[f], in B 2. IfAis uniformly elliptic, [[A]] BMO(B2 ) 1, and u W 1,2 (B 2 ) is a weak solution, then ( ) p/2 u p dx C(n, p) F p dx + u 2 dx. B 1 B2 B 2 This is first proved by Calderón-Zygmund witha C(B 2 ), then extended by many mathematicians toa BMO: F. Chiarenza, M. Frasca, and P. Longo, (1991); L.A. Caffarelli and I. Peral (1998); L. Wang - S-.S. Byun (200s); Krylov (200s),...
Remarks on the homogeneity in the CZ theory The equations div[a(x) u] = div[f], in B 2 is invariant under the dilation: If u is a solution, then u λ (x) := u(λx)/λ is also a solution div[a λ (x) u λ ] = div[f λ ], in B 2/λ. wherea λ (x) =A(λx), F λ (x) = F(λx). The estimate ( u p dx C(n, p) F p dx + B 1 B2 ) p/2 u 2 dx B 2 is also invariant under this dilation. = The CZ reflects the true homogeneity of the PDEs. See also from the Hardy-Littilewood maximal functions.
Difficulties in our questions For our equation { div[a(x, u, u)] = div[f(x)] in Ω, u = g(x) on Ω, It is non-uniformly elliptic becuasea(x, z,η) µ(x)η with µ A 2. The homogeneity of the PDE is broken due to the dependent ofaon u. The CZ is not expected in this case by experts. Note: However, in caseais independent on u, the PDE is still invariant with the dilation. The CZ is available in this case indeed. We would love to have two weighted estimates: Weak solution u W 1,2 (Ω,µ) and the data F/µ L p (Ω,ω), g W 1,p (Ω,ω), with another weightω. Hope to obtain u L p (Ω,ω)?
Asymptotically Uhlenbeck vector fields Definition A : Ω R R n Ris asymptotically Uhlenbeck if there is a symmetric measurable matrixã:ω R n n, and a bounded continuous functionω 0 :K [0, ) [0, ) such that [ ] A(x, z,η) Ã(x)η ω 0 (z, η ) 1+ η µ(x), for for all most every x Ω, (z,η) R R n, and lim ω 0(z, s) = 0, uniformly in z, for z K. s A prototypical example is A(x, z,η) = a(x, z, η )η, with lim s a(x, z, s) = ã(x), uniformly in z R and Ã(x) = ã(x)i n.
Sawyer s condition Definition Letµ,ω be any two positive, locally finite Borel measures onr n and let 1<p<. The pair of measures (µ,ω) is said to satisfy the p-sawyer s condition if there is a constant C> 0 such that ˆ ( dσ ) pdω Cσ(B), M µ (χ B dµ ) ball B R n, B whereχ B is the characteristic function of the ball B, dσ = ( ) p dµ/dω dω, and 1 p + 1 p = 1.
Sawyer two weighted estimates for Hardy-littlewood maximal function Theorem (E. R. Sawyer - 1982)) Letµ,ω be any two positive, locally finite Borel measures onr n and let 1<q<. Then, M L µ q (R n,ω) L q (R n,ω) C, if and only if the pair (µ,ω) satisfies the q-sawyer s condition, where M µ f(x) = sup f(y) d(µ(y)). ρ>0 B ρ (x) See also I. Verbitsky (1992), and D. Cruz-Uribe (2000).
Our main results where Ω R (y 0 ) = Ω B R (y 0 ) and y 0 Ω, (µ,ω) is the p 2 -Sawyer s pair. Theorem (T. P. 2017) Assume thata(x, z,η) is degenerate asµ(x)η andais asymptotically Uhlenbeck. Assume also that [[Ã]] BMO(Ω,µ) 1, and Ω is sufficiently flat. If u W 1,p (Ω,µ) is a weak solution of { div[a(x, u, u)] = div[f(x)] in Ω, u = g(x) on Ω, there is a constant C = C(p, q,λ, M 0, M 1, M 2,ω 0, n)>0such that the estimate ˆ [ˆ ˆ u p ω(x)dx C g p ω(x)dx + F/µ p ω(x)dx Ω R (y 0 ) Ω 2R (y 0 ) Ω 2R (y 0 ) ( ˆ p/2 1 +ω(ω R (y 0 )) u µ(x)dx) 2 + 1 µ(b 2R (y 0 )) Ω 2R (y 0 )
Ideas and main steps the proof Clearly, the main task is to deal with the inhomogeneity of the nonlinear PDE. Enlarge the class of equation to study: Forλ>0, study the class of equations with parameterλ { div[aλ (x, u, u)] = div[f(x)] in Ω, u = g(x) on Ω, wherea λ (x, z,η) =A(x,λz,λη)/λ. This class of nonlinear degenerate PDEs are invariant under the dilation: u s (x) = u(sx)/s Establish the CZ theory for the above equation whenλ λ 0, with someλ 0 =λ 0 (Λ, n,ω) sufficiently large. Use scaling argument to obtain the CZ estimate forλ=1.
Conclusion We study regularity estimate of Calderón-Zygmund type for nonlinear equations of Uhlenbeck type, with non-homogeneous boundary condition. Two weighted nonlinear Calderón-Zygmund type regularity estimates are established. Results are new, even for the uniformly elliptic case, because of the dependent of the nonlinearity of the equations on solutions. Linear equations, one weighted Calderón-Zygmund type regularity estimates are established earlier, by D. Cao - T. Mengesha - T. P. (2016).
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