SUBELLIPTIC CORDES ESTIMATES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S XX SUBELLIPTIC CORDES ESTIMATES Abstract. We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W, - regularity for p-harmonic functions in the Heisenberg group H for the range 7 p < Introduction The main goal of this paper is to prove some estimates of Cordes type for subelliptic partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group, including the linearized p-laplacian. To show the applicability of our methods let us state the following theorem that constitutes a special case of our results. Theorem.. Let 7 p < Then any p-harmonic function in the Heisenberg group H initially in HW,p, loc is in HWloc. We build on previous regularity results obtained by Marchi [7, 8] and extended by the first author [3], which give non-uniform bounds of the HW, or HW,p norm of the approximate p-harmonic functions. Using the Cordes condition [, ] and Strichartz s spectral analysis [0] we establish HW, estimates for linear subelliptic partial differential operators with measurable coefficients. As an application we obtain uniform HW, bounds for the approximate p-harmonic functions for p in a range that depends on the dimension of the Heisenberg group H n. Consider the Heisenberg group H n ; that is R n+ with the group multiplication x,..., x n, t y,..., y n, u = x +y,..., x n +y n, t+u n x i y n+i x n+i y i. For i {,..., n} consider the vector fields X i = x n+i x i t, X n+i = + x i x n+i i= t, T = t. The nontrivial commutators are [X i, X n+i ] = T, otherwise [X i, X j ] = 0. Let H n be a domain. Consider the following Sobolev space with respect to the horizontal vector fields X i as HW, = {u L : X i X j u L, for all i, j {,..., n}} 000 Mathematics Subject Classification. 35H0, 35J70. Key words and phrases. Cordes conditions, subelliptic equations, p-laplacian. The authors were partially supported by NSF award DMS c 997 American Mathematical Society

2 endowed with the inner-product u, v HW, = uxvx + i,j= X i X j ux X i X j vx dx. HW, is a Hilbert space and let HW, 0 be the closure of C0 in this Hilbert space. We denote by X u the matrix of second order horizontal derivatives whose entries are X u ij = X j X i u, and by H u = n i= X ix i u the subelliptic Laplacian associated to the horizontal vector fields X i. Lemma.. For all u HW, 0 we have where X u L c n H u L, c n = The constant c n is sharp when = H n. + n. Proof. We follow the spectral analysis of H developed by Strichartz [0]. Let us recall the fact that H and it commute, and share the same system of eigenvectors Φ λ,k,l z, t = λ n π n+ n + k exp λ z 4n + k where l = ±, k {0,,,...} and L n k exp n+ L n k ilλt n + k λ z, n + k is the Laguerre polynomial L n k t = et t n k! dk dt k e t t k+n. For the eigenvalues, we have the following relations it u Φ λ,k,l = lλ n + k u Φ λ,k,l. H u Φ λ,k,l = λu Φ λ,k,l,. where denotes the group convolution. Therefore, the spectral decomposition of H u for u C0, the Plancherel formula, and relations.-. give H u L = π n + k H u Φ λ,k,l z, 0 dzdλ k=0 l=± 0 C n = π n + k n + k it u Φ λ,k,l z, 0 l dzdλ k=0 l=± n T u L Therefore, for all u C 0 we have 0 C n T u L n Hu L..3

3 CORDES CONDITIONS 3 In the following we will use the fact that the formal adjoint of X k is X k. Let u C0. For k {,..., n} and j k + n, X k and X j commute, therefore X k X j ux dx = X k X k ux X j X j uxdx. For j = k + n we have X k X j ux dx = X k X j ux X j X k ux + T ux dx = X k X j ux X j X k ux dx + X k X j ux T ux dx = X j ux X k X j X k ux dx + X k X j ux T ux dx = X j ux X j X k + T X k ux dx + X k X j ux T ux dx = X j ux X j X k X k ux dx + X k X j ux T ux dx = X k X k ux X j X j ux dx + X k X j ux T ux dx. Similarly, Therefore, = = = X u L = k,j= The constant T v = i n Hv. k= X j X k ux dx X k X k ux X j X j ux dx k,j= X k X j u L = k= X j X k ux T ux dx. n X k X k ux X j X j ux dx + [X k, X k+n ]ux T ux dx X k X k ux dx + n + n n H u L = + n T ux dx H u L. + n is sharp when = Hn, because for v = Φ λ,0, we have. Cordes conditions for second order subelliptic PDE operators in non-divergence forms with measurable coefficients Let us consider now Au = i,j= a ij xx i X j u

4 4 where the functions a ij L. Let us denote by A = a ij the n n matrix of coefficients. Definition.. [, ] We say that A satisfies the Cordes condition K ε,σ if there exists ε 0, ] and σ > 0 such that 0 < σ i,j= a ijx n + ε i= a ii x, a.e. x.. Theorem.. Let 0 < ε, σ > 0 such that γ = ε c n < and A satisfies the Cordes condition K ε,σ. Then for all u HW, 0 we have X u L + n γ α L Au L,. where αx = Ax, I Ax. Proof. We denote by I the identity n n matrix, by A, B = n i,j= a ijb ij n the inner product and by A = i,j= a ij the Euclidean norm in Rn n for matrices A and B. The Cordes condition K ε,σ implies that Ax, I n ε.3 Ax for all x, where the Lebesgue measure of \ is 0. Let be now x arbitrary, but fixed. Consider the quadratic polynomial Inequality.3 shows that Therefore there exists P α = Ax α Ax, I α + n ε. min P α = P α R αx = such that P αx 0. Observing that Ax, I Ax 0..4 Ax, I Ax.5 I αxax = Ax α x Ax, I αx + n we get that.4 implies that which is equivalent to I αxax ε, I αxax, M ε M, for all M M n R..6 Condition.6 can be written also as n n m ii αx a ij xm ij ε i= i,j= n i,j= m ij /.7

5 CORDES CONDITIONS 5 for all M M n R. Formula.7 and Lemma. imply that for all u HW, 0 we have H ux αxaux dx ε X i X j ux dx εc n Therefore, for γ = ε c n < we get i,j= H ux dx. H u αau L γ H u L which shows that X u L c n H u L c n γ αau L c n γ α L Au L. 3. HW, -interior regularity for p-harmonic functions in H n Let H n be a domain, h HW,p and p >. Consider the problem of minimizing the functional Φu = Xux p dx over all u HW,p such that u h HW,p 0. The Euler equation for this problem is the p-laplace equation X i Xu p X i u = 0, in. 3. i= A function u HW,p is called a weak solution for 3. if Xux p X i ux X i ϕxdx = 0, ϕ HW,p i= Φ is a convex functional on HW,p, therefore weak solutions are minimizers for Φ and vice-versa. For m N let us define now the approximating problems of minimizing functionals Φ m u = m + Xux and the corresponding Euler equations p X i m + Xu X i u i= i= p = 0, in. 3.3 The weak form of this equation is p m + Xux X i ux X i ϕxdx = 0, for all ϕ HW,p

6 6 The differentiated version of equation 3.3 has the form where i,j= a m ij X i X j u = 0, in 3.5 a m ij x = δ ij + p X iux X j ux m +. Xux Let us consider a weak solution u m HW,p of equation 3.3. Then a m ij L. Define the mapping L m : W, 0 L by and L m vx = i,j= a m ij xx i X j vx. 3.6 We will check the validity of Theorem. for L m. We have i,j= i= a m Xu m ii x = n + p m + Xu m, a m ij x Xu m = n + p m + Xu m + p Xu m 4 m + Xu m. Denote Xu m p m + Xu m = Λ. Therefore, for an ε c, we need n n + Λ + Λ n + ε n + Λ. This leads to Hence, n Λ ε n + Λ + Λ < < n + Λ + Λ c. n n c n Λ Λ n < 0. Solving this inequality we get n n c Λ n + n c, + n n c n + n n c. 3.7 n Using c n = n+ n have p and the fact that Xu m m + Xum < we have that for all m N we n n 4n + 4n 3 n, n + n 4n + 4n 3 + n n + n, 3.8 and that the operators L m satisfies the assumptions of Theorem. uniformly in m.

7 CORDES CONDITIONS 7 Let us remark that in the case n = we have 5 p, + 5. Theorem 3.. Let p < + n + n 4n + 4n 3 n + n If u HW,p is a minimizer for the functional Φ, then u HW, loc. Proof. The case p = it is well known, so let us suppose p. Let u HW,p be a minimizer for Φ. Consider x 0 and r > 0 such that B 4r = Bx 0, 4r. We need a cut-off function η C0 B r such that η = on B r. Also consider minimizers u m for Φ m on HW,p B r subject to u m u HW,p 0 B r. Then u m u in HW,p B r as m. By [3, 7] we get that for p < 4 we have u m HW, loc, but with bounds depending on m, and also that u m satisfies equation L m u m = 0 a.e. in B r. So, in B r we have a.e. X i X j η u m = X i X j η u m + X j η X i u m + X i η X j u m + η X i X j u m and hence L m η u m = u m L m,um η + By Theorem. it follows that i,j=. a m ij x X j η X i u m + X i η X j u m. X u m L B r X η u m L B r c L m η u m L B r c u m HW,p B r c u HW,p B r where c is independent of m. Therefore, u HW, B r. Remark 3.. Observe that the range for p given by Theorem 3. is shrinking from [, 5+ 5 to [, 3] as n increases from to. For the case p < we need the following lemmas. The first lemma is an interpolation result and its proof is based on integration by parts. Lemma 3.. For all u C 0 and for all δ > 0 there exists cδ > 0 such that Proof. Xu L = = δ i= Xu L δ X u L + cδ u L. X i ux X i ux dx = i= ux H ux dx δ H ux dx + cδ n X ux dx + cδ u x dx ux X i X i uxdx = u x dx

8 8 From Lemma 3. and the higher order extension results available for the Sobolev spaces on the Heisenberg group [6, 9] we get the following result. Lemma 3.. For all u HW, B r and all δ > 0 there exists cδ > 0 such that Xu L B r δ X u L B r + cδ u L B r. By Lemmas 3. and 3. we can use a method similar to the proof of Theorem 9. [5] to get the following result. Lemma 3.3. Let us suppose that the operator A satisfies the assumptions of Theorem 4. and that B 3r. Then X u L B r c Au L B r + u L B r, for all u HW, loc B 3r. Proof. Let η C0 B r, 0 < σ < and σ = +σ such that η is a cut-off function between B σr and B σ r satisfying Xη σr and 4 X η σ r. Then we can use Theorem. for ηu to get X u L B σr X ηu L B r c Aηu L B r = c ηau + u Aη + a ij X u j ηx i u + X i ηx j c Au L B r + i,j= σr Xu L B σ r + L B r σ r u L B σ r For k {0,, } let us use the seminorms Then u k = sup σ k r k X k u L B σr. 0<σ< u c r Au L B r + u + u 0. Lemma 3. implies that for δ > 0 small we have Therefore, and hence For σ = X u L B σr u δ u + cδ u 0. u c r Au L B r + u 0 we get the desired inequality. c σ r r Au L B r + u L B r. Theorem 3.. Let us consider the Heisenberg group H and 7 p. If u HW,p is a minimizer for the functional Φ, then u HW, loc.

9 CORDES CONDITIONS 9 Proof. We start the proof in the same way as we did in the proof of Theorem 3.. Let u HW,p be a minimizer for Φ. Consider x 0 and r > 0 such that B 4r = Bx 0, 4r. We need a test function η C0 B 3r. Also consider minimizers u m for Φ m on HW,p B 3r subject to u m u HW,p 0 B 3r. Then u m u in HW,p B 3r as m. We use the facts that 4 3 < < <, that the homogeneous dimension of H is Q = 4, and 4p for all 4 4 p 3 p <. The Sobolev embeddings result in the subelliptic setting [] says that HW,p 0 B3r L q 4p B 3r, for q 4 p. Therefore, u m u in L B 3r. Also, using that see [3] for 7 p we have u m HW,p loc B3r we get that Xum L loc B3r. Let us remark that these bounds of X u m in L p may depend on m and that L m u m = 0 a.e. in B 3r. Moreover, L m η u m L B 3r = = c u m L m η + a m,u ij x X j η X i u m + X i η X j u m i,j= L B 3r c u m L suppη + Xu m L suppη < +. and hence u m HW, loc B3r. By Lemma 3.3 for all m sufficiently large we have X u m L B r c u m L B r c u L B r which shows that X u m is uniformly bounded in HW, B r, hence u HW, B r. In the forthcoming article [4] we establish the C,α regularity for p-harmonic functions in H n when p is in a neighborhood of. Acknowledgement. The authors would like to thank Robert S. Strichartz for his valuable hint regarding the spectral analysis of the sublaplacian and estimate.3, and Silvana Marchi for making available her papers and for her conversations with the second author. References [] L. Capogna, D. Danielli and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations, 8993, [] H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proceedings of Symposia in Pure Mathematics IV 96. [3] A. Domokos, Fractional difference quotients along vector fields and applications to second order differentiability of solutions to quasilinear subelliptic equations in the Heisenberg group, submitted for publication. [4] A. Domokos and Juan J. Manfredi, C,α -regularity for p-harmonic functions in the Heisenberg group for p near, submitted for publication.

10 0 [5] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 983. [6] G. Lu, Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups, Acta Math. Sinica, English Series, 6000, [7] S. Marchi, C,α local regularity for the solutions of the p-laplacian on the Heisenberg group for p + 5, Journal for Analysis and its Applications 000, See Erratum, to appear. [8] S. Marchi, C,α local regularity for the solutions of the p-laplacian on the Heisenberg group. The case + < p, Comment. Math. Univ. Carolinae 44003, See Erratum, 5 to appear. [9] D. Nhieu, Extension of Sobolev spaces on the Heisenberg group, C. R. Acad. Sci. Ser. I Math., 3995, [0] R. S. Strichartz, L p Harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Analysis, 9699, [] G. Talenti, Sopra una classe di equazioni ellitiche a coeficienti misurabili, Ann. Mat. Pura. Appl , Department of Mathematics, University of Pittsburgh, 30 Thackeray Hall, 560 Pittsburgh, PA,USA address: and36@pitt.edu Department of Mathematics, University of Pittsburgh, 30 Thackeray Hall, 560 Pittsburgh, PA,USA address: manfredi@pitt.edu