Electronic Journal of Differential Equations, Vol. 27 27), No. 2, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR ELLIPTIC EQUATIONS YONGYONG WANG, JUNJIE ZHANG, SHENZHOU ZHENG Communicated by Zhasheng Feng Abstract. We prove a global Lorentz estimate of the Hessian of strong solutions to a class of asymptotically regular fully nonlinear elliptic equations over a C, smooth bounded domain. Here, the approach of the main proof is based on the Possion s transform from an asymptotically regular elliptic equation to the regular one.. Introduction Let Ω be a bounded domain with Ω C, in R n for n 2. The main purpose of this paper is to attain a global Calderón-Zygmund type estimate in the scale of Lorentz spaces for the Hessian of strong solutions to the Dirichlet problem of asymptotical regular fully nonlinear elliptic equations of nondivergence form. The studied problem is F x, D 2 u) = f in Ω,.) u = on Ω, where the real valued function F x, D 2 u) : Ω Sn) R is an asymptotically regular elliptic operator which Sn) is the space of real n n symmetric matrices, and f is any given function in Lorentz spaces L γ,q Ω) with γ > n and < q. The Calderón-Zygmund estimate is a popular research to various elliptic and parabolic problems in recent decades. In the settings of discontinuous coefficients, an interior and boundary W 2,p estimate for linear elliptic equations with VMO Vanishing Mean Oscillations) discontinuous coefficients was first proved by Chiarenza, Frasca, and Longo [8, 9]. Since then, there have been a lot of research activities on the Calderón-Zygmund theory of elliptic and parabolic equations and systems in divergence or non-divergence form. Regarding fully nonlinear elliptic equations.), an interior W 2,p estimate was first obtained by Caffarelli in [6] if f L p with p > n under the assumption of a small measure to the oscillation of F x, M) in the variable x uniformly for M. Later, Caffarelli and Huang [7] further showed that under the assumptions of F x, M) with a small multiplier of BMO in x and the prerequisite of Evans-Krylov estimates [2], if f belongs to the generalized Campamato- John-Nirenberg spaces, then D 2 u correspondingly belongs to the same spaces as f. 2 Mathematics Subject Classification. 35J6, 35B65, 35D3. Key words and phrases. Nonlinear elliptic equations; Lorentz estimate; asymptotically regular; small BMO coefficients. c 27 Texas State University. Submitted October 25, 26. Published May 4, 27.
2 Y. WANG, J. ZHANG, S. ZHENG EJDE-27/2 Recently, Winter [2] also used a similar technique to establish the corresponding boundary estimate so as to get a global W 2,p -solvability of the associated boundaryvalue problem. All these papers showed that a small oscillation assumption in the L or L n integral average sense is imposed on the operators F x, M) in x. Recently, Krylov et al [, 4, 5] developed W 2,p -solvability for fully nonlinear elliptic and parabolic equations with VMO coefficients whose local oscillations are measured in a certain average sense allowing rather rough discontinuity. After that, Dong-Krylov-Li [] demonstrated an interior solvability in W 2,p for p > n and W 2, p for p > n +, respectively, to fully nonlinear elliptic and parabolic equations with VMO coefficients in bounded domains or cylinders. Moreover, Byun et al [4] also attained the global weighted W 2,p estimates of the Hessian for fully nonlinear elliptic equations with small BMO coefficients in a bounded C, domain via rather different geometrical approaches. On the other hand, Lorentz spaces are a two-parameter scale of spaces which refine Lebesgue spaces in some sense. It is an important observation that requiring the principle coefficients to have small mean oscillations in the integral average sense is sufficient to achieve higher integrability and Lorentz regularity. Since the pioneering work of Talenti [2] based on symmetrization, there were a large of literature on the topic of Lorentz regularity to elliptic and parabolic PDEs. Recently, Mengesha-Phuc in [6] used a kind of geometrical approach to prove the weighted Lorentz regularity of the gradient for quasilinear elliptic p-laplacian equations, and Zhang-Zhou [24] extended their results to the setting of quasilinear px)-laplacian. Meanwhile, Baroni in [, 2] made use of so-called Large-M-inequality principle introduced by Acerbi-Mingione to show the Lorentz estimates of gradient for evolutionary p-laplacian systems and obstacle parabolic p-laplacian, respectively. We would like to point out that another key ingredient is that F x, D 2 u) is assumed to be an asymptotically regular. It was Chipot and Evans [] to first introduce the notion of asymptotically regular in the elliptic framework, and Raymond [7] further considered a Lipschitz regularity to asymptotically regular problems with p-growth. Since then there are a large of literatures on the topic of asymptotically regular problems. In particular, Scheven and Schmidt in [8, 9] recently obtained a local higher integrability and a local partial Lipschitz continuity with a singular set of small positive measure for the gradient Du to the system which exhibits a certain kind of elliptic behavior near infinity, respectively. Furthermore, a global Lipschitz regularity result was extended by Foss in [3]. Very recently, Byun-Oh-Wang [5] proved global Calderón-Zygmund estimates for nonhomogeneous asymptotically regular elliptic and parabolic problems in divergence form in the Reifenberg flat domain by covering the given asymptotically regular problems to suitable regular problems. Furthermore, Byun-Cho-Oh [3] extended the same conclusions to the setting of nonlinear obstacle elliptic problems. Zhang- Zheng [22, 23] also further extended the work of Byun-Oh-Wang [5] to the case of obstacle parabolic problems in the scale of Lorentz spaces. Inspired by those recent works mentioned above, in this paper we consider a global Lorentz estimate of the Hessian of strong solutions to the Dirichlet problem.) for asymptotical regular fully nonlinear elliptic equations over a C, bounded domain. More precisely, our aim is to attain a global Lorentz estimate of the second derivative to the Dirichlet problem.) with asymptotically regular nonlinearity. Indeed, it is a natural refined outgrowth of Byun et al s recent papers [5]. In
EJDE-27/2 LORENTZ ESTIMATES 3 particular, the Lebesgue space L γ is a special case of Lorentz space L γ,q when q = γ. Before stating the main result, let us give some basic concepts and facts. Let us first recall that the Lorentz space L γ,q Ω) with γ < and < q <, which is the set of measurable function g : Ω R such that q/γ g q dµ L γ,q Ω) := q µ γ {ξ Ω : gξ) > µ} ) µ < +. While the Lorentz space L γ, for γ <, q = is set to be the usual Marcinkiewicz space M γ Ω) with quasinorm ) g L γ, = g Mγ Ω) := sup µ γ γ {ξ Ω : gξ) > µ} < +. µ> The local variant of such spaces is defined in the usual way. Moreover, we note that by Fubini s theorem there holds ) g γ dµ L γ Ω) = γ µ γ {ξ Ω : gξ) > µ} µ = g γ L γ,γ Ω), so that L γ Ω) = L γ,γ Ω); cf. [, 2]. In this context, we denote by Cn, λ, Λ,... ) a universal constant depending only on prescribed quantities and possibly varying from line to line. In this article, we are interested in the case that F x, M) is asymptotically elliptic. This is to say that it is getting closer to some real-valued function Gx, M) as M goes to infinity, where Gx, M) satisfies the following uniformly elliptic assumption. Definition.. uniformly ellipticity) We say Gx, M) : Ω Sn) R is uniformly elliptic if there exist constants < λ Λ < such that for any x Ω and any M Sn), there holds λ N Gx, M + N) Gx, M) Λ N, N..2) Remark.2. i) We write N whenever N is a non-negative definite symmetric matrix. N denotes the L 2, L 2 )-norm of N, that is, N = sup x = Nx. Therefore N is equal to the maximum eigenvalue of N whenever N. ii) The uniformly elliptic assumption implies that Gx, M) is monotone increasing and Lipschitz in M Sn). We next introduce the definition of asymptotically elliptic operators. Definition.3. F x, M) is asymptotically elliptic if there exists a uniformly elliptic operator Gx, M) and a bounded function ω : R + R + with lim ωr) = r such that for all M Sn) and any x Ω. F x, M) Gx, M) ω M ) + M ).3) By a direct calculation, we conclude that uniformly with respect to x Ω. F x, M) Gx, M) lim =,.4) M M
4 Y. WANG, J. ZHANG, S. ZHENG EJDE-27/2 We define function β G to measure the oscillation of Gx, M) in the variable x. Let G : Ω Sn) R and let x Ω be fixed. For x Ω, we define β G x, x ) := sup M Sn)\{} Now, let us summarize our main results as follows. Gx, M) Gx, M)..5) M Theorem.4. Assume γ > n and < q. Let u W 2,n Ω) be a strong solution to.) with f L γ,q Ω) and Ω C,. Then there exists a small positive constant β = β n, λ, Λ, γ, q) such that if F x, D 2 u) is asymptotically elliptic with Gx, D 2 u) satisfying uniformly elliptic condition and B r x ) Ω β G x, x ) n dx) β,.6) for any x Ω and < r < R. We suppose that Gx, D 2 u) is convex and positive homogeneous of degree one in D 2 u. Then we have D 2 u L γ,q Ω), satisfying the estimate D 2 u L γ,q Ω) C f L γ,q Ω) + ),.7) where C = Cn, λ, Λ, γ, q, Ω). In the case q = the constant C depends only on n, λ, Λ, γ, Ω. To realize our aim, some ideas from [5] are employed in our main proof. For example, to get the global Lorentz estimate we use an equivalent representation of Lorentz norm, the Hardy-Littlewood maximal functions, and the Poisson formula by constructing a regular problem from the given irregular problem. The rest of this article is organized as follows. In section 2, we first prove the global Lorentz estimates of the corresponding regular problem, and then we give a proof of the main result by taking a transformation from given asymptotically regular problem to a suitable regular problem. 2. Proof of Theorem.4 We prove Theorem.4 by employing an appropriate transformation to construct a uniformly elliptic operator from a given asymptotically elliptic operator. To this end, we assume that real-valued function F x, M) is asymptotical to a uniformly elliptic Gx, M), which satisfies convex and positive homogeneous of degree one in M and β G x, x ) dx) n β, for any x Ω and < r < R, where β > will be determined later. By Definition.3, we have F x, M) Gx, M) lim =. M M Now we write a real-valued function Hx, M) : Ω Sn) by M Hx, M) := F x, M) Gx, M); 2.) then there exists an K = Kβ ) > such that M K Hx, M) β, x Ω. 2.2)
EJDE-27/2 LORENTZ ESTIMATES 5 For any fixed point x Ω, we next define a new real-valued function Hx, M) by Hx, M) if M K, M Hx, M) := K H K x, M M) if < M < K, 2.3) if M =. It follows that Hx, M) is also convex in M, positive homogeneous of degree one in M, and Hx, M) β, M Sn), 2.4) uniformly with respect to x Ω. Note that Hx, M) = Hx, M) if M K. Therefore, for M we have F x, M) = Gx, M) + M Hx, M) = Gx, M) + M Hx, M) + M Hx, M) Hx, M)) = Gx, M) + M Hx, M) + M χ{m Sn): M <K} Hx, M) Hx, M)), 2.5) where χ{m Sn) : M < K} denotes the characteristic function on the set {M Sn) : M < K}. In the setting of M =, we define M Hx, M) M= := F x, ) Gx, ), then the formula 2.5) still holds for all M Sn). Let u W 2,n be a strong solution of the Dirichlet problem.). Define G : Ω Sn) R by Then, by 2.5) and 2.6), it yields Gx, M) := Gx, M) + M Hx, D 2 u). 2.6) F x, D 2 u) = Gx, D 2 u) + D 2 u χ{ D 2 u <K}Hx, D 2 u) Hx, D 2 u)), 2.7) where χ{ D 2 u < K} = χ{x Ω : D 2 u < K} denotes the characteristic function on the set {x Ω : D 2 ux) < K}. Thus, from.) it implies that u is a strong solution of Gx, D 2 u) = f + D 2 u χ{ D2 u <K} Hx, D 2 u) Hx, D 2 u)) := g, x Ω. 2.8) To prove Theorem.4, we also need to show that the new nonlinearity G satisfies uniformly ellipticity and the oscillation condition.6) in the L n integral average sense with small constant 3β. More precisely, we have the following lemma. Lemma 2.. Let u W 2,n Ω) be a strong solution of the Dirichlet problem.). Assume that F x, M) is asymptotically elliptic with Gx, M) satisfying β G x, x ) dx) n β, 2.9) for any x Ω and < r < R. Then we have the following conclusions: i) If < β λ/3, then Gx, M) is uniformly elliptic. ii) For any x Ω and < r < R, Gx, M) satisfies β Gx, x ) dx) n 3β. 2.)
6 Y. WANG, J. ZHANG, S. ZHENG EJDE-27/2 Proof. i) Let < β λ 3. For any M, N Sn) with N, we have Gx, M + N) Gx, M) = Gx, M + N) Gx, M) + M + N M ) Hx, M) because of 2.6). From 2.4) it follows that Hx, D 2 u) β, 2.) uniformly with respect to x Ω. We have the triangle inequality Hence, by.2), 2.) and 2.2), we find that and M + N M N. 2.2) Gx, M + N) Gx, M) λ N β N = λ β ) N 2λ 3 N, Gx, M + N) Gx, M) Λ N + β N = Λ + β ) N Λ + λ 3 ) N, since < β λ 3. Namely, λ N Gx, M + N) Gx, M) Λ N, 2.3) where λ = 2 3 λ and Λ = Λ + λ 3. So the assertion i) is proved. ii) Let x Ω and < r < R. For any x B r x ) Ω, it follows from 2.4) and 2.6) that which implies Gx, M + N) Gx, M) Gx, M) Gx, M) + 2β M, 2.4) β Gx, x ) = sup Gx, M) Gx, M) M Sn)\{} M Gx, M) Gx, M) sup + 2β M Sn)\{} M = β G x, x ) + 2β. Therefore, by 2.9), 2.5) and the Minkowski inequality we obtain ) β Gx, x ) n dx B r x ) Ω β G x, x ) dx) n + 2β B r x ) Ω β + 2β = 3β, which implies the assertion ii). 2.5) 2.6) We recall an interior Lorentz estimate of strong solutions to fully nonlinear uniformly elliptic equations, whose proof can be found in [23] by using the approach of large-m-inequality principle originated from Acerbi-Mingione s work. More precisely, let us consider the fully nonlinear uniformly elliptic equations with f L γ,q Ω). Gx, D 2 u) = fx), in Ω 2.7)
EJDE-27/2 LORENTZ ESTIMATES 7 Lemma 2.2 [23, Corollary.4]). Assume γ > n and < q. Let u W 2,n Ω) be a strong solution to 2.7) satisfying uniformly ellipticity and f L γ,q Ω). If Gx, D 2 u) is convex and positive homogeneous of degree one in D 2 u, then there exists a small positive β = β n, λ, Λ, γ, q) such that if Gx, D 2 u) satisfies B r x ) Ω β G x, x ) n dx) β, for every x Ω and < r < R ; then we have D 2 u L γ,q loc Ω). Moreover, there exists a radii R = R n, λ, Λ, γ, q) such that for each ball B 2R x ) Ω and < R R with the estimate D 2 u L γ,q B R ) C D 2 u L n B 2R ) + f L γ,q B 2R )), 2.8) where C = Cn, λ, Λ, γ, q). In the case q = the constant C and R above depend only on n, λ, Λ, γ. Next, we establish a local boundary estimate in the scale of Lorentz spaces by using the idea of odd/even extensions over the flat boundary. Fixed a point x Ω, without loss of generality let us write Ω is flat near x lying in the plane {x = }. Then we may assume there exists an open ball B 2R x ) with center x and radius 2R such that We also set Γ 2R = B 2R x ) {x = }. B + 2R := B 2Rx ) {x > } Ω, B 2R := B 2Rx ) {x < } R n Ω. Lemma 2.3. For γ > n and < q, let u W 2,n Ω) be a strong solution of local boundary value problem Gx, D 2 u) = fx), in B + 2R, u =, on Γ 2R 2.9) with f L γ,q Ω). Then there exist small positive constants δ and R depending only on n, λ, Λ, γ, q such that, Gx, D 2 u) satisfying uniformly elliptic, Gx, D 2 u) is convex and positive homogeneous of degree one in D 2 u and β G x, x ) dx) n β, we have D 2 u L γ,q B + R ) C D 2 u L n B + 2R ) + f L γ,q B + 2R ) ), 2.2) for each half ball B + 2R with < R R, where C = Cn, λ, Λ, γ, q). In the case q =, the constant C above depends only on n, λ, Λ, γ. Proof. Note that Gx, D 2 u) is convex and positive homogeneous of degree one in D 2 u and β G x, x ) dx) n β, which implies Gx, D 2 u) = G D 2 ij ux, D 2 u)d 2 iju := a ij x)d 2 iju.
8 Y. WANG, J. ZHANG, S. ZHENG EJDE-27/2 Let us now define û in B 2R x ) with x on the flat boundary by { ûx, x ux, x ) if x, ) = u x, x ) if x <, and extend a ij x) = a ij x, x ) from {x } to {x < } by even or odd reflection, depending on the indices i and j. Specifically, when x, â ij x) = a ij x); when x <, { a ij x, x ), if i = j = or i, j {2,..., n}, â ij x) = a ij x, x ), if i {2,..., n} and j =. Also set â j = â j. We see that the nonlinearity Ĝx, D2 û) satisfy uniformly elliptic, convex and positive homogeneous of degree one in D 2 û and β bg x, x ) dx) n β, Let ˆf be the odd extension of f with respect to x, then it is easy to check that ˆf L γ,q B 2R x )). By Lemma 2.2 it implies that the extended û is a strong solution of Ĝx, D2 û) = ˆf in B 2R x ), then it gives rise to the local Lorentz estimate 2.8). Therefore, the desired estimate 2.2) is obtained by restricting û from B 2R x ) to B + 2R. Using the standard flattening and covering arguments, we can derive a global Lorentz estimate as follows. Theorem 2.4. For γ > n and < q, let u W 2,n Ω) be a strong solution to the following Dirichlet problem Gx, D 2 u) = fx), in Ω, u =, on Ω, 2.2) Gx, D 2 u) satisfying uniformly elliptic, convex and positive homogeneous of degree one in D 2 u with the following oscillation on coefficients β G x, x ) dx) n β for some small positive constant β = β n, λ, Λ, γ, q). If f L γ,q Ω) and Ω C,, then D 2 u L γ,q Ω) and there exists a positive constant C = Cn, λ, Λ, γ, q, Ω) with the estimate D 2 u L γ,q Ω) C f L γ,q Ω). 2.22) While q =, the constant C depends only on n, λ, Λ, γ, Ω. Proof. ) For fixed any point x Ω, we now flatten the boundary near x in order to apply the flat boundary estimates 2.2). Thanks to the assumption Ω C,, there exists a neighborhood N x and a C, -diffeomorphism Φ : N B 2R such that Φx ) =, and ΦN Ω) = B + 2R ). We write y = Φx), x N Ω, and define Ψ = Φ, then x = Ψy). Define ũy) = uψy)) = ux) for y B + 2R. Then it is readily checked that ũ W 2,n B + 2R ) is a strong solution of the flat initial-boundary problem F y, D 2 ũ) = fy) in B + 2R,
EJDE-27/2 LORENTZ ESTIMATES 9 where ũ = on Γ 2R B + 2R, F y, D 2 ũ) := F Ψy), DΦ T Ψ ) D 2 ũ DΦ Ψ) + Dũ D 2 Φ Ψ )), fy) := fψy)). It is obvious that F is convex in D 2 ũ and F y, ) =. Moreover, we readily see that β ef y, y ) CΦ)β F Ψy), Ψy )) for any y, y B + 2R ; and F satisfies the similar assumptions of Lemma 2.3 with different positive constants. Therefore, it yields D 2 ũ L γ,q B + R ) C D 2 ũ L n B + 2R ) + f ) L γ,q B + 2R ). Converting back to the original x-variables, we conclude D 2 u L γ,q ΨB + R )) C D 2 u Ln ΨB + 2R )) + f L γ,q ΨB + 2R )) ), 2.23) From this estimate, along with the interior bound 2.8) in Lemma 2.2, the standard covering arguments lead to D 2 u L γ,q Ω) C D 2 u L n Ω) + f L γ,q Ω)), 2.24) for some positive constant C depending on n, λ, Λ, γ, q, Ω, which implies u W 2 L γ,q Ω) C Du L γ,q Ω) + D 2 u L n Ω) + f L γ,q Ω)). 2.25) 2) At this point, the desired estimate 2.22) follows from the uniqueness property of the homogeneous equation. Indeed, if 2.22) is not true, there exists a sequence {u k } k= and {f k} k= such that u k for each k is a strong solution of the problem with the estimate F x, D 2 u k ) = f k x) in Ω, u k = on Ω, u k W 2 L γ,q Ω) > k f k L γ,q Ω), for all k. 2.26) Without loss of generality, we may suppose that Then it follows from 2.26) that u k W 2 L γ,q Ω) =. 2.27) f k L γ,q Ω) < k, as k. 2.28) Since {u k } k= is uniformly bounded in W 2 L γ,q Ω), there exists a subsequence, which be still denoted by {u k } k=, and a function u W 2 L γ,q Ω), such that u k u weakly in W 2 L γ,q Ω), u k u in L γ,q Ω), as k. 2.29) It is easy to check that u is a strong solution of F x, D 2 u ) =, in Ω, u =, on Ω. 2.3)
Y. WANG, J. ZHANG, S. ZHENG EJDE-27/2 Accordingly, u = due to the uniqueness of strong solutions to zero initialboundary problem 2.3), so it follows from 2.28) and 2.29) that u k f k u k in L γ,q Ω), weakly in W 2 L γ,q Ω), in L γ,q Ω), as k. Note that W 2 L γ,q Ω) W 2 L n Ω) because γ > n, hence 2.3) u k L n Ω), Du k L n Ω), as k. 2.32) Moreover, letting the measure ν = dx, we see that which implies Du k ν-a.e. in Ω as k up to subsequence), {x Ω : Du k > µ} for all µ > as k, so by the Lebesgue Dominated Convergence Theorem we obtain Du k in L γ,q Ω) as k. 2.33) Combining 2.25), 2.27), 2.3), 2.32) and 2.33), it yields C Du k L γ,q Ω) + D 2 u k Ln Ω) + f k L γ,q Ω)) as k, which is a contradiction. This completes the proof. In view of the lemma 2., the asymptotically elliptic equation.) turns out to be a uniformly elliptic equation 2.8). For the asymptotically elliptic equation, lemma 2. and the existing theory for fully nonlinear, uniformly elliptic equation, see Lemma 2.2, will be employed to finally derive the required estimate. We are now ready to prove our main result. Proof of theorem.4. From 2.8), for any given positive constant β, as in Lemma 2.2, we define a new data β := min{ λ 3, }, and set β := 3 min{β, β } >. Then there exists a real-valued function Hx, D 2 u) such that Hx, D 2 u) β < for all x Ω. Now let u W 2,p be a strong solution of the problem.) and assume that F x, D 2 u) is asymptotically elliptic with Gx, D 2 u) which satisfies β G x, x ) dx) n β, B r x ) Ω for any x Ω and < r < R. Then by Lemma 2.2, Gx, D 2 u) is uniformly elliptic and satisfies β Gx, x ) dx) n 3 β β, B r x ) Ω for any x Ω and < r < R. According to f L γ,q Ω) and equality 2.8), we have gx) fx) + 2 D 2 ux) χ{ D 2 u <K}.
EJDE-27/2 LORENTZ ESTIMATES which implies that Therefore, Since {x Ω : gx) > µ } {x Ω : fx) > µ 2 } + {x Ω : 2 D2 ux) χ{ D 2 u <K} > µ 2 }. g q L γ,q Ω) q + q = 2 q q we can derive that Thus, we have + 2 q q µ γ {x Ω : fx) > µ ) q/γ dµ 2 } µ µ γ {x Ω : 2 D 2 ux) χ { D2 u <K} > µ ) q/γ dµ 2 } µ µ γ {x Ω : fx) > µ} ) q/γ dµ µ µ γ {x Ω : 2 D 2 ux) χ { D 2 u <K} > µ} ) q/γ dµ µ. {x Ω : 2 Du 2 x) χ { D2 u <K} > µ} {x Ω : 2K > µ}, g q L γ,q Ω) 2q f q L γ,q Ω) + 2q q = 2 q f q L γ,q Ω) + 2q q + 2 q q 2K 2 q f q L γ,q Ω) + 2q q 2K µ γ {x Ω : 2K > µ} ) q/γ dµ µ µ γ {x Ω : 2K > µ} ) q/γ dµ µ µ γ {x Ω : 2K > µ} ) q/γ dµ µ 2K µ γ Ω ) q/γ dµ µ + 2K = 2 q f q L γ,q Ω) + 2q q Ω q/γ µ q dµ = 2 q f q L γ,q Ω) + 2q Ω q/γ 2K) q C f q L γ,q Ω) + ). g L γ,q Ω) C f L γ,q Ω) + ), for some positive constant C = Cβ, K, n, γ, q, Ω ). Considering F x, M) Gx, M) Hx, M) = M for all M Sn) and x Ω, we have Hx, M) for all x Ω due to the definition of H. This shows that Hx, D 2 u) for all x Ω. On the other hand, we know Gx, M) and M are convex in M; therefore F x, M) = Gx, M)+ Hx, D 2 u) M is also convex with respect to M. We then apply Theorem 2.4 to g L γ,q Ω) and Gx, D 2 u) to discover u W 2,n with the estimate D 2 u L γ,q Ω) C g L γ,q Ω) C f L γ,q Ω) + ), 2.34) where C = Cn, λ, Λ, R, Ω, γ, q) is a positive constant. This completes the proof.
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EJDE-27/2 LORENTZ ESTIMATES 3 Yongyong Wang Department of Mathematics, Beijing Jiaotong University, Beijing 44, China E-mail address: yongyongwang@bjtu.edu.cn Junjie Zhang Department of Mathematics, Beijing Jiaotong University, Beijing 44, China E-mail address: junjiezhang@bjtu.edu.cn Shenzhou Zheng Corresponding author) Department of Mathematics, Beijing Jiaotong University, Beijing 44, China E-mail address: shzhzheng@bjtu.edu.cn