We investigate the regularity of transmission problems for a general class of uniformly elliptic fully non linear equations. We prove that, if the forcing term is Lipschitz, then viscosity solution are C1,\u3b3

de Silva, Daniela

Ferrari, Fausto

Salsa, Sandro

English

We investigate the regularity of transmission problems for a general class of uniformly elliptic fully non linear equations. We prove that, if the forcing term is Lipschitz, then viscosity solution are C1,γ

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Two nonlinear days in Urbino 2017,Electronic Journal of Differential Equations, Conference 25 (2018), pp. 55–63.ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.eduREGULARITY OF TRANSMISSION PROBLEMS FORUNIFORMLY ELLIPTIC FULLY NONLINEAR EQUATIONSDANIELA DE SILVA, FAUSTO FERRARI, SANDRO SALSADedicated to our loved Anna AloeAbstract. We investigate the regularity of transmission problems for a gen-eral class of uniformly elliptic fully non linear equations. We prove that, if theforcing term is Lipschitz, then viscosity solution are C1,γ .1. Introduction and main resultsIn this article we consider viscosity solutions to the transmission problemF+(D2u) = f+ in B+1 := B1(0) ∩ {xn > 0}F−(D2u) = f− in B−1 := B1(0) ∩ {xn < 0}a(un)+ − b(un)− = 0 on B′1 := B1(0) ∩ {xn = 0},(1.1)where (un)+ and (un)− denote the derivatives in the en direction of u restrictedto the upper and lower half ball, respectively. Here a > 0, b ≥ 0 and F± are fullynonlinear uniformly elliptic operators, with ellipticity constants Λ ≥ λ > 0 andF±(0) = 0. That is, for every M,N ∈ Sn,M−λ,Λ(N) ≤ F±(M +N)−F±(M) ≤M+λ,Λ(N),where Sn denotes the set of square symmetric n×n matrices andM−λ,Λ,M+λ,Λ arethe extremal Pucci operators defined byM−λ,Λ(N) = infA∈Aλ,ΛTr(AN), M+λ,Λ(N) = supA∈Aλ,ΛTr(AN),Aλ,Λ = {A ∈ Sn : λI ≤ A ≤ ΛI}.In the sequel we write F±(D2u) = f±, in B±1 to denote both interior equations in(1.1). Now we define viscosity solution for problem (1.1).Definition 1.1. We say that u ∈ C(B1) is a viscosity subsolution (supersolution)to (1.1) if(i) F±(D2u) ≥ f± (≤ f±) in B±1 in the viscosity sense;2010 Mathematics Subject Classification. 35J60, 35B65.Key words and phrases. Transmission problems; fully nonlinear equations;regularity of solutions.c©2018 Texas State University.Published September 15, 2018.5556 D. DE SILVA, F. FERRARI, S. SALSA EJDE-2018/CONF/25(ii) Let x0 ∈ {xn = 0}, δ > 0 small, and ϕ ∈ C2(B+δ (x0)) ∩ C(B−δ (x0)). If ϕtouches u from above (below) at x0, thena(ϕn)+(x0)− b(ϕn)−(x0) ≥ 0 (≤ 0).We say that u is a viscosity solution if it is both a viscosity subsolution andsupersolution.It is easily seen that condition (ii) in Definition 1.1 can be replaced by thefollowing condition(ii’) Letψ(x) = P (x′) + px+n − qx−n ,where P is a quadratic polynomial. If ψ touches from above (below) u atx0 ∈ {xn = 0}, thenap− bq ≥ 0 (≤ 0).Transmission problems as (1.1) play a key role for example in the regularity theoryfor two-phase free boundary problems developed by the authors in [2, 3, 4]. Ourpurpose here is to review and extend to the case of distributed sources the regularityresults about transmission problems provided in [4] for the homogeneous case. Ourmain result is the following one.Theorem 1.2. Let u be a viscosity solution to (1.1) in B1. Assume that f± ∈C0,1(B±1 ) ∩ L∞(B1). Then, for any ρ < 1, u ∈ C1,α(B̄±ρ ) with norm bounded bya constant depending on n, λ,Λ, ρ, ‖u‖∞, ‖f‖∞ and ‖f±‖C0,1 . In particular thetransmission condition is satisfied in the classical sense.2. Hölder continuityIn this section we prove the Hölder continuity of a solution u to problem (1.1).Here we only need f± ∈ C(B±1 )∩L∞(B1). We introduce a special class of functions,based on the extremal Pucci operators, in the spirit of [1]. Since a > 0 and b ≥ 0 aredefined up to a multiplicative constant and the problem is invariant under reflectionwith respect to {xn = 0}, we can assume that a = 1, 0 ≤ b ≤ 1.We denote by Sλ,Λ(f±) the class of continuous functions u in B1 such thatM+λ,Λ(D2u) ≥ f± in B±1 , and(un)+ − b(un)− ≥ 0 on B′1in the sense of Definition 1.1 with comparison functions touching u from above.Analogously, we denote by Sλ,Λ(f±) the class of continuous functions u in B1such thatM−λ,Λ(D2u) ≤ f± in B±1 , and(un)+ − b(un)− ≤ 0 on B′1in the sense of Definition 1.1 with comparison functions touching u from below.Finally we setS∗λ,Λ(f±) = Sλ,Λ(−|f±|) ∩ Sλ,Λ(|f±|).Theorem 2.1. Let u ∈ S∗λ,Λ(f±) with f± ∈ C(B±1 )∩L∞(B1). Then u ∈ Cα(B1/2)with α and ‖u‖Cα(B1/2) depending on n, λ,Λ, ‖f±‖∞ and ‖u‖∞.EJDE-2018/CONF/25 REGULARITY IN TRANSMISSION PROBLEMS 57It is sufficient to show that u ∈ Cα(Bρ0) with ρ0 small depending only on ‖f±‖∞.Then after the rescaling u(x)→ u(rx), the theorem follows easily from the iterationof the following lemma. Here constants depending on n, λ,Λ are called universal.Lemma 2.2. Let u ∈ S∗λ,Λ(f±) with ‖u‖∞ ≤ 1 and ‖f±‖∞ ≤ ε0 in B1, ε0 smalluniversal. Assume that at x̄ = 15enu(x̄) > 0. (2.1)Then u ≥ −1 + c in B1/3 with 0 < c < 1 universal.Proof. By Harnack inequality, if ε0 is small enough depending on the Harnackconstant, and by assumption (2.1), we deduce that (c0 universal)u ≥ −1 + c0 in B1/20(x̄). (2.2)Let r = |x− x̄| andw = η(Γγ(r) + δx+n ) +ε02λx2n,Γγ(r) = r−γ − (2/3)−γ , δ = −12Γγ(34)be defined in the ring D = B3/4(x̄)\B1/20(x̄), with γ > max{0, Λλ (n− 1)− 1} andη to be chosen later. Since Γγ is a radial function, in an appropriate system ofcoordinatesD2w = ηγr−γ−2diag{(γ + 1),−1, . . . ,−1}.Hence, in D,M−λ,Λ(D2w) ≥ η(γr−γ−2(λ(γ + 1)− Λ(n− 1))) + ε0 ≥ ‖f±‖∞. (2.3)Since ∂nΓγ > 0 on {xn = 0}, the transmission condition(wn)+ − b(wn)− > 0 on {xn = 0}, (2.4)is satisfied. Now choose η, ε0 small universal so thatw ≤ c0 on ∂B1/20(x̄).Then, by choosing ε0 possibly smaller, we also have w ≤ 0 on ∂B3/4(x̄). Thus, inview of (2.2) we obtain thatw ≤ u+ 1 on ∂D.From (2.3), (2.4) w is a strict classical subsolution of the transmission problemfor the operator M−λ,Λ with right hand sides |f±|. By the definition of viscositysolution, we conclude that it must bew ≤ u+ 1 in D.Since w ≥ c in B1/3 the proof is complete. 3. Upper and lower envelopesGiven a continuous function u in B1(0) and Bρ ⊂ B1(0) we define for ε > 0 theupper ε-envelope of u in the x′-direction,uε(y′, yn) = supx∈Bρ∩{xn=yn}{u(x′, yn)−1ε|x′ − y′|2} y = (y′, yn) ∈ Bρ.Note that there is yε ∈ Bρ ∩ {xn = yn} such thatuε(y) = u(yε)−1ε|y′ε − y′|258 D. DE SILVA, F. FERRARI, S. SALSA EJDE-2018/CONF/25with |y′ − y′ε| ≤√2ε‖u‖∞, since uε(y) ≥ u(y) and1ε|y′ε − y′|2 = u(yε)− uε(y) ≤ u(yε)− u(y).Lemma 3.1. The following properties hold:(1) uε ∈ C(Bρ) and uε ↓ u uniformly in Bρ as ε→ 0.(2) uε is C1,1 in the x′-direction by below in Bρ. Thus uε is pointwise secondorder differentiable in the x′-direction al almost every point of Bρ.(3) If u is a viscosity solution to (1.1), then, in a smaller ball Br, r ≤ ρ −2√ε‖u‖∞, uε is a viscosity subsolution toF±(D2uε) = f± − ωf±(√2ε‖u‖∞) in B±r(uεn)+ − b(uεn)− = 0 on B′r(3.1)where ωf± denotes the modulus of continuity of f±.Proof. (1) follows from|uε(y0)− uε(y1)| ≤6ρε|y′0 − y′1|and0 ≤ uε(y)− u(y) = u(yε)−1ε|y′ε − y′|2 − u(y) ≤ ωu(√2ε‖u‖∞)where ωu is the modulus of continuity of u.(2) follows from Alexandrov Theorem on concave/convex functions, sinceuε(y′, yn) +1ε|y′|2 = supx∈Bρ∩{xn=yn}{u(x′, yn)−1ε|x′|2 + 2ε〈x′, y′〉}is convex, being the supremum of a family of affine functions of y′.(3) Let ϕ ∈ C2(Br) touch from above uε at a point x̄ ∈ B+r . Then (ε small)uε(x̄) = u(x̄ε)−1ε|x̄′ε − x̄′|2with |x̄′ε − x̄′|2 ≤ 2ε‖u‖∞. Consider the functionΦ(y) = ϕ(y + x̄− x̄ε) +1ε|x̄′ε − x̄′ε|2.With our choice of r and y ∈ B+ρ close enough to x̄ε, the point y + x̄ − x̄ε ∈ B+ρ .Thus, by the definition of uε,u(y) ≤ uε(y + x̄− x̄ε) +1ε|x̄′ − x̄′ε|2and thereforeu(y) ≤ ϕ(y + x̄− x̄ε) +1ε|x̄′ − x̄′ε|2.with equality at y = x̄ε, since ϕ(x̄) = uε(x̄). Thus the function Φ touches fromabove u at x̄ε. ThereforeF+(D2ϕ(x̄)) = F+(D2Φ(x̄ε)) ≥ f+(xε) ≥ f+(x̄)− ωf+big(√2ε‖u‖∞).Similarly, we can check the transmission condition(uεn)+ − b(uεn)− ≥ 0 on B′r.Letϕ(x) = P (x′) + px+n − qx−nEJDE-2018/CONF/25 REGULARITY IN TRANSMISSION PROBLEMS 59touch from above uε at a point x̄ = (x̄′, 0) ∈ B′r, with P quadratic polynomial.ThenP (x′) + px+n − qx−n ≥ uε(x)near x̄ andP (x̄′) = uε(x̄) = u(x̄ε)−1ε|x̄′ − x̄′ε|2.Then, for r small and y close enough to x̄εu(y) ≤ uε(y + x̄− x̄ε) +1ε|x̄′ − x̄′ε|2≤ ϕ(y + x̄− x̄ε) +1ε|x̄′ε − x̄′ε|2with equality at y = x̄ε since P (x̄′) = uε(x̄). Hence ϕ(y + x̄ − x̄ε) + 1ε |x̄′ε − x̄′ε|2touches from above u at y = x̄ε and therefore p− bq ≥ 0, as desired. Analogously we can defineuε(y′, yn) = infx∈Bρ∩{xn=yn}{u(x′, yn) +1ε|x′ − y′|2}y = (y′, yn) ∈ Bρ.the lower ε-envelope of u in the x′-direction. Properties (1)–(3) hold with obviouschanges:(1’) uε ∈ C(Bρ) and uε ↑ u uniformly in Bρ as ε→ 0.(2’) uε is C1,1 in the x′-direction by above in Bρ. Thus uε is pointwise secondorder differentiable in the x′-direction al almost every point of Bρ.(3’) If u is a viscosity solution to (1.1), then, in a smaller ball Br, r ≤ ρ −2√ε‖u‖∞, uε is a viscosity supersolution toF±(D2uε) = f± + ωf±(√2ε‖u‖∞) in B±r((uε)n)+ − b((uε)n)− = 0 on B′r.(3.2)4. Proof of Theorem 1.2For the proof of Theorem 1.2 we use the following pointwise regularity result(see [5]).Theorem 4.1. Let F be a uniformly elliptic operator and u be a viscosity solutionto the Dirichlet problemF(D2u) = f in B+1 , u(x′, 0) = g(x′) on B′1.Assume that g is pointwise C1,α at 0 and f ∈ L∞(B+1 ). Then u is poinwise C1,αat 0, that is there exists a linear function Lu such that for all r small|u− Lu| ≤ Cr1+α in B̄+rwith C depending only on n, λ,Λ, ‖f‖∞ and the pointwise C1,α bound of g at 0.Our main lemma reads as follows.Lemma 4.2. Let f± ∈ C0,1(B±1 ) and u be a viscosity solution to (1.1) in B1.Then, for any σ > 0 small, and any unit vector e′ in the x′ direction,u(x+ σe′)− u(x) ∈ S∗λ,Λ(ωf±(σ)) .60 D. DE SILVA, F. FERRARI, S. SALSA EJDE-2018/CONF/25Proof. Let v = u(x+σe′), w = v−u. Following the proof of [1, Theorem 5.3], withminor modification, it follows thatM+λ,Λ(D2w) ≥ −ωf±(σ) and M−λ,Λ(D2w) ≤ ωf±(σ)in B±1 .To show that the free boundary condition is satisfied in the viscosity sense, weslightly modify the technique in [4] for the homogeneous case.ϕ(x) = P (x′) + px+n − qx−n − Cx2ntouch w from above at x0 = (x′0, 0), where P is a quadratic polynomial. Choosingsuitably C and possibly restricting the neighborhood around x0, we can assumethatM+λ,Λ(D2ϕ) ≤ −2ωf±(σ)− 2ωf±(√2‖u‖∞). (4.1)Assume by contradiction thatp− bq < 0.Let δ > 0 small, and consider the ring Aδ(x0) = B̄2δ(x0)\Bδ(x0). Without loss ofgenerality we can assume that ϕ touches w strictly from above and therefore thatϕ − w ≥ η > 0 on Aδ(x0). Since wε = uε − vε converges uniformly to u − v, forε small enough (up to adding a small constant), we have that ϕ touches wε fromabove at some point xε and ϕ− wε ≥ η/2 on ∂Bδ(xε). In view of (4.1), we haveM+λ,Λ(D2wε) ≥ −ωf±(σ)− 2ωf±(√2ε‖u‖∞) >M+λ,Λ(D2ϕ)and therefore xε ∈ {xn = 0}. Callψ = ϕ− wε − η/2.Since ψ ≥ 0 on ∂Bδ(xε), ψ(xε) < 0 and ϕ touches wε from above at xε, from ABPestimates ([1, Lemma 3.5]) it follows that the set of points in Bδ(xε) ∩ {xn = 0}where ψ admits a touching plane l(x′) from below, in the x′ direction, of slope lessthan some arbitrary small number has positive measure. We choose the slope ofl small enough to have that ϕ̄ = ϕ − l − η/2 ≥ wε on ∂Bδ(xε) and hence in theinterior. By property (2) in Lemma 3.1 we can deduce that ϕ̄(yε) = wε(yε) forsome yε = (y′ε, 0) where both vε and uε are twice pointwise differentiable in the x′direction.Now we call ūε the solution toF±(D2ūε(x)) = f±(x)− ωf±(√2ε‖u‖∞) in B±δ (xε)with w = uε on ∂Bδ(xε), and similarly let v̄ε the solution toF±(D2v̄ε(x)) = f±(x+ σe′) + ωf±(√2ε‖u‖∞) in B±δ (xε)with w = vε on ∂Bδ(xε), Setw̄ε = ūε − v̄ε.Then M+λ,Λ(D2w̄ε) ≥ −ωf±(σ) − 2ωf±(√2ε‖u‖∞) >M+λ,Λ(D2ϕ̄) in Bδ(xε) withϕ̄ ≥ wε on the boundary. Thus, by comparison, we deduce that ϕ̄ ≥ w̄ε ≥ wε alsoin the interior with contact from above at yε.Also note that, by comparison, the replacements ūε and v̄ε are sub and supersolution of the transmission condition on {xn = 0}, respectively.EJDE-2018/CONF/25 REGULARITY IN TRANSMISSION PROBLEMS 61By the pointwise C1,α differentiability at yε of the boundary data, we concludethe C1,α differentiability of v̄ε and ūε up to yε. Thus there are linear functionsLv, Lu such that, for all small r,|v̄ε − Lv| ≤ Cr1+α,|ūε − Lu| ≤ Cr1+αin B+r (yε). Since ϕ̄ touches w̄ε from above, we get p ≥ p+u − p+v , wherep+u = (ūε)+n (yε), p+v = (v̄ε)+n (yε).Arguing similarly in B−r (yε) we infer q ≤ q−u − q−v , whereq−u = (ūε)−n (yε), q−v = (v̄ε)−n (yε).In the next lemma we show thatp+u − p+v − b(q−u − q−v ) ≥ 0thus reaching a contradiction. Lemma 4.3. Let g± ∈ C(B±1 ) and u be a viscosity solution to (0 ≤ b ≤ 1)F±(D2u) = g± in B±1(un)+ − b(un)− ≥ 0 (≤ 0) on B′1.(4.2)Assume that u is twice differentiable at x = 0 in the x′-direction. Then u is differ-entiable at 0 andu+n (0)− bu−n (0) ≥ 0 (≤ 0).Proof. From Theorem 4.1 there exists a linear function Lu such that, for all smallr,|u− Lu| ≤ Cr1+α in B+r .Without loss of generality, by subtracting a linear function, we may assume thatLu = (un)+(0)xn := d+xn.Let w be the solution toF+(D2w) = g+ in B+r , w = ϕr on ∂B+rwhereϕr ={2C|x|1+α on ∂B+r ∩ {xn > 0}2Crα−1|x′|2 on B′r.Since u is twice pointwise differentiable at 0, we get, for r small enough,u− d+xn ≤ ϕr on ∂B+rand, by comparison,u− d+xn ≤ w in B+r . (4.3)Now, the rescaling Wr(x) = r−1−αw(rx) solvesG(D2Wr) = r1−αg+(rx) in B+1 , Wr(x′) = 2C|x′|2 on B′1where G(M) = r1−αF+(rα−1M) has the same ellipticity constants of F+.By boundary C1,α estimates we obtain that‖Wr‖C1,α(B+1/2) ≤ C̄62 D. DE SILVA, F. FERRARI, S. SALSA EJDE-2018/CONF/25for a universal C̄. In particularWr(x) ≤ 2C|x′|2 + C̄xn in B̄+1/2.Rescaling back we getw(x) ≤ 2Crα−1|x′|2 + C̄rαxn in B̄+r/2.From (4.3),u ≤ 2Crα−1|x′|2 + C̄rαxn + d+xn in B+r/2.Arguing similarly in B−r we findu ≤ 2Crα−1|x′|2 + C̄rαxn + d−xn in B−r/2with d− = (un)+(0). Thusϕ(x) = 2Crα−1|x′|2 + (C̄rα + d+)x+n − (−C̄rα + d−)x−ntouches u by above at zero. Therefore(C̄rα + d+)− b(−C̄rα + d−) ≥ 0for all small r so that d+ − bd− ≥ 0. From Theorem 4.1 and the arguments in Chapter 5 in [1] we deduce the followingresult.Corollary 4.4. Let u be a viscosity solution to (1.1). Then u ∈ C1,α in the x′-direction in B3/4 with norm bounded by a constant depending on n, λ,Λ, ‖u‖∞ and‖f±‖C0,1 .We are now ready to give the proof of our main Theorem 1.2.Proof of Theorem 1.2. Let, say ρ = 1/2. The C1,α regularity and the bounds on‖u‖C1,α(B1/2) follow from Corollary 4.4 and the regularity theory for fully nonlinearuniformly elliptic equations in [5] or [6]. It remains to show that the transmissioncondition is satisfied in the classical sense. Let us prove that at x = 0, ap− bq ≤ 0where p = (un)+(0), q = (un)−(0). By the C1,α regularity of u, after possiblysubtracting the linear function u(0) +∇x′u(0) · x′, we can write|u(x)− (px+n − qx−n )| ≤ Cr1+α |x| ≤ r. (4.4)For r small, definewr(x) = Crα−1(−|x|2 +Kx2n)− 2rαCK|xn|+ px+n − qx−n .Choose K large to haveM−λ,Λ(D2wr) ≥ C(2λ(K − 1)− 2Λn) > ‖f±‖∞. (4.5)Using (4.4), we get wr < u on ∂Br. Letm = minBr(u− wr) = (u− wr)(x0).Since M−λ,Λ(D2u) ≤ F±(D2u) = f±, from (4.5) we deduce that x0 /∈ B±r . Also,since (u − wr)(0) = 0 it follows that m ≤ 0, hence x0 /∈ ∂Br. Thus x0 ∈ B′r andwr +m touches u at x0 from below. By definition it follows thata(p− 2rαCK)− b(q + 2rαCK) ≤ 0.Letting r → 0 we get ap− bq ≤ 0. EJDE-2018/CONF/25 REGULARITY IN TRANSMISSION PROBLEMS 63Acknowledgments. F. Ferrari was supported by INDAM-GNAMPA 2017: Rego-larità delle soluzioni viscose per equazioni a derivate parziali non lineari degeneri.References[1] L. A. Caffarelli, X. Cabré; Fully nonlinear elliptic equations, American Mathematical SocietyColloquium Publications, 43. American Mathematical Society, Providence, RI, 1995.[2] D. De Silva, F. Ferrari, S. Salsa; Two-phase problems with distributed source: regularity ofthe free boundary, Anal. PDE, 7 (2014), no. 2, 267–310.[3] D. De Silva, F. Ferrari, S. Salsa; Perron’ssolutions for two-phase free boundary problems withdistributed sources, Nonlinear Anal., 121 (2015), 382–402.[4] D. De Silva, F. Ferrari, S. Salsa; Free boundary regularity for fully nonlinear non-homogeneoustwo-phase problems, J. Math. Pures Appl., (9) 103 (2015), no. 3, 658–694.[5] F. Ma, L. Wang; Boundary first order derivative estimates for fully nonlinear elliptic equa-tions, J. Differential Equations, 252 (2012), no. 2, 988–1002.[6] E. Milakis, L.E. Silvestre; Regularity for fully nonlinear elliptic equations with Neumannboundary data, Comm. Partial Differential Equations, 31 (2006), no. 7-9, 1227–1252Daniela De SilvaDepartment of Mathematics, Barnard College, Columbia University, New York, NY10027, USAE-mail address: desilva@math.columbia.eduFausto FerrariDipartimento di Matematica dell’Università di Bologna, Piazza di Porta S. Donato, 5,40126 Bologna, ItalyE-mail address: fausto.ferrari@unibo.itSandro SalsaDipartimento di Matematica del Politecnico di Milano, Piazza Leonardo da Vinci, 32,20133 Milano, ItalyE-mail address: sandro.salsa@polimi.it

Regularity of transmission problems for uniformly elliptic fully nonlinear equations

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Regularity of transmission problems for uniformly elliptic fully nonlinear equations

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