On the regularity theory for mixed local and nonlocal quasilinear elliptic equations
aa r X i v : . [ m a t h . A P ] F e b On the regularity theory for mixed local andnonlocal quasilinear elliptic equations
Prashanta Garain and Juha KinnunenMarch 1, 2021
Abstract
We consider a combination of local and nonlocal p -Laplace equations and discuss sev-eral regularity properties of weak solutions. More precisely, we establish local boundednessof weak subsolutions, local H¨older continuity of weak solutions, Harnack inequality forweak solutions and weak Harnack inequality for weak supersolutions. We also discusslower semicontinuity of weak supersolutions as well as upper semicontinuity of weak sub-solutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Mosertheory, the expansion of positivity and estimates involving a tail term. The main re-sults apply to sign changing solutions and capture both local and nonlocal features of theequation.Keywords: Regularity, mixed local and nonlocal p -Laplace equation, local boundedness,H¨older continuity, Harnack inequality, weak Harnack inequality, lower semicontinuity,energy estimates, De Giorgi-Nash-Moser theory, expansion of positivity. In this article, we develop regularity theory of weak solutions for the problem − ∆ p u + L ( u ) = 0 in Ω , < p < ∞ , (1.1)where Ω is a bounded domain in R n . The local p -Laplace operator is defined by∆ p u = div( |∇ u | p − ∇ u ) , (1.2)and L is the nonlocal p -Laplace operator given by L ( u )( x ) = P.V. ˆ R n | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) K ( x, y ) dy, (1.3)where P.V. denotes the principal value. Here K is a symmetric kernel in x and y such thatΛ − | x − y | n + ps ≤ K ( x, y ) ≤ Λ | x − y | n + ps , (1.4)1 ixed local and nonlocal quasilinear equations ≥ < s <
1. Note that the p -Laplace operator ∆ p reduces to the classicalLaplace operator ∆ for p = 2. When K ( x, y ) = | x − y | − ( n + ps ) , the operator L is the fractional p -Laplace operator ( − ∆ p ) s , which further reduces to the fractional Laplacian ( − ∆) s for p = 2.A prototype problem of type (1.1) is − ∆ p u + ( − ∆ p ) s u = 0 , < p < ∞ , < s < . (1.5)Before describing our contribution, let us discuss some of the known results. In the localcase, the regularity theory for the p -Laplace equation ∆ p u = 0 has been studied extensively,for example, see Lindqvist [28], Mal´y-Ziemer [29] and references therein. For the nonlocal p -Laplace equation ( − ∆ p ) s u = 0 , (1.6)a scale invariant Harnack inequality holds for globally nonnegative solutions, but fails whenthe solution changes sign as shown by Kassmann [24] in the case p = 2. These results havebeen extended to 1 < p < ∞ by Di Castro-Kuusi-Palatucci [13]. In addition, a weak Harnackinequality for supersolutions of (1.6) is obtained in [13]. They introduced a nonlocal tailterm to compensate for the sign change in the Harnack estimates. Di Castro-Kuusi-Palatucci[14] studied local boundedness estimate along with H¨older continuity of solutions for (1.6).See also Brasco-Lindgren-Scikorra [7] for higher regularity results. For lower semicontinuityresults of supersolutions, we refer to Korvenp¨a¨a-Kuusi-Lindgren [26].For the mixed local and nonlocal case with p = 2, i.e. − ∆ u + ( − ∆) s u = 0 , (1.7)Foondun [22] have proved Harnack inequality and local H¨older continuity for nonnegative solu-tions. Barlow-Bass-Chen-Kassmann [3] have obtained a Harnack inequality for the parabolicproblem related to (1.7). Chen-Kumagai [12] have proved Harnack inequality and local H¨oldercontinuity for the parabolic problem of (1.7). Such a parabolic Harnack estimate has beenused to prove elliptic Harnack inequality for (1.7) by Chen-Kim-Song-Vondraˇcek in [11]. Formore regularity results related to (1.7), we refer the reader to Athreya-Ramachandran [1],Chen-Kim-Song [9] and Chen-Kim-Song-Vondraˇcek [10]. The arguments in these articlescombine probability and analysis. Moreover, the Harnack inequality is proved only for glob-ally nonnegative solutions. Recently, an interior Sobolev regularity, a strong maximum prin-ciple and a symmetry property among many other qualitative properties of solutions for (1.7)have been studied by Biagi-Dipierro-Valdinoci-Vecchi [4, 5], Dipierro-Proietti Lippi-Valdinoci[17, 18] and Dipierro-Ros-Oton-Serra-Valdinoci [19] using an analytic approach.We establish the following regularity results for weak solutions (Definition 2.5) of (1.1)with 1 < p < ∞ and 0 < s < • Local boundedness of weak subsolutions (Theorem 4.2). The argument is based onan energy estimate (Lemma 3.1), the Sobolev inequality and an iteration technique(Lemma 4.1). ixed local and nonlocal quasilinear equations • Local H¨older continuity of weak solutions (Theorem 5.1). Local H¨older continuity isnot direct consequence of the Harnack inequality in the nonlocal case, see [3, 21]. Wefollow the approach of Di Castro-Kuusi-Palatucci [14] in which the local boundednessestimate and the logarithmic energy estimate (Lemma 3.4) play an important role. • Harnack inequality (Theorem 8.2) for weak solutions and weak Harnack inequality forweak supersolutions (Theorem 8.3). The expansion of positivity (Lemma 7.1), the localboundedness result and a tail estimate (Lemma 6.1) are crucial here. • Lower and upper semicontinuity of weak supersolutions and subsolutions, respectively(Theorem 9.2 and Corollary 9.3). This result is an adaptation to the mixed local andnonlocal case of a measure theoretic approach (Lemma 9.1) in Liao [27]. We refer toBanerjee-Garain-Kinnunen [2] for an adaptation of this approach to a class of doublynonlinear parabolic nonlocal problems.In contrast to the techniques from probability and analysis introduced in [1, 3, 9, 10, 11,12, 22], our approach is purely analytic and based on the De Giorgi-Nash-Moser theory. Tothe best of our knowledge, all of our main results are new for p = 2. Moreover, some of ourmain results (Theorem 4.2, Theorem 8.3, Theorem 9.2 and Corollary 9.3) seem to be new evenfor p = 2. Furthermore, our approach applies to sign changing solutions. In this respect, ourHarnack estimate (Theorem 8.2) also extends the result of Chen-Kim-Song-Vondraˇcek [11]and Foondun [22] to sign changing solutions. We introduce a tail term (Definition 3.2), thatdiffers from the one discussed in [13], and a tail estimate (Lemma 6.1) that capture both localand nonlocal features of (1.1). Technical novelties include an adaptation of the expansion ofpositivity technique (Lemma 7.1) for the mixed problem.This article is organized as follows. In Section 2, we discuss some definitions and pre-liminary results. Necessary energy estimates are proved in Section 3. In Sections 4 and 5,we establish the local boundedness and H¨older continuity results. In Sections 6 and 7, weobtain a tail estimate and the expansion of positivity property. In Section 8, we prove Har-nack and weak Harnack estimates. Finally, in Section 9, we establish the lower and uppersemicontinuity results. In this section, we present some known results for fractional Sobolev spaces, see Di Nezza-Palatucci-Valdinoci [15] for more details.
Definition 2.1
Let < p < ∞ and < s < . Assume that Ω ⊂ R n be any domain in R n .The fractional Sobolev space W s,p (Ω) is defined by W s,p (Ω) = (cid:26) u ∈ L p (Ω) : ˆ Ω ˆ Ω | u ( x ) − u ( y ) | p | x − y | n + ps dx dy < ∞ (cid:27) ixed local and nonlocal quasilinear equations and endowed with the norm k u k W s,p (Ω) = (cid:18) ˆ Ω | u ( x ) | p dx + ˆ Ω ˆ Ω | u ( x ) − u ( y ) | p | x − y | n + ps dx dy (cid:19) p . The fractional Sobolev space with zero boundary value is defined by W s,p (Ω) = (cid:8) u ∈ W s,p ( R n ) : u = 0 on R n \ Ω (cid:9) . Both W s,p (Ω) and W s,p (Ω) are reflexive Banach spaces, see [15]. The space W s,p loc (Ω) isdefined by requiring that a function belongs to W s,p (Ω ′ ) for every Ω ′ ⋐ Ω. Here Ω ′ ⋐ Ωdenotes that Ω ′ is a compact subset of Ω. Throughout, we write c or C to denote a constantwhich may vary from line to line or even in the same line. If c and C depends on r , r , . . . , r k we write c = c ( r , r , . . . , r k ) and C = C ( r , r , . . . , r k ) respectively.The next result asserts that the standard Sobolev space is continuously embedded inthe fractional Sobolev space, see [15, Proposition 2.2]. The argument applies a smoothnessproperty of Ω so that we can extend functions from W ,p (Ω) to W ,p ( R n ) and that theextension operator is bounded. Lemma 2.2
Let Ω be a smooth bounded domain in R n , < p < ∞ and < s < . Thereexists a constant C = C ( n, p, s ) such that || u || W s,p (Ω) ≤ C || u || W ,p (Ω) for every u ∈ W ,p (Ω) . The following result for the fractional Sobolev spaces with zero boundary value followsfrom [8, Lemma 2.1]. The main difference compared to Lemma 2.2 is that the result holdsfor any bounded domain, since for the Sobolev spaces with zero boundary value, we alwayshave zero extension to the complement.
Lemma 2.3
Let Ω be a bounded domain in R n , < p < ∞ and < s < . There exists aconstant C = C ( n, p, s, Ω) such that ˆ R n ˆ R n | u ( x ) − u ( y ) | p | x − y | n + ps dx dy ≤ C ˆ Ω |∇ u | p dx for every u ∈ W ,p (Ω) . Here we consider the zero extension of u to the complement of Ω . The following version of the Gagliardo-Nirenberg-Sobolev inequality will be useful for us,see [29, Corollary 1.57].
Lemma 2.4
Let < p < ∞ and Ω ⊂ R n be an open set with | Ω | < ∞ . Assume κ = nn − p , if < p < n, , if p ≥ n. (2.1) ixed local and nonlocal quasilinear equations Then for any u ∈ W ,p (Ω) , there exists a positive constant C = C ( n, p ) such that (cid:18) ˆ Ω | u | κp dx (cid:19) κp ≤ C | Ω | n − p + κp (cid:18) ˆ Ω |∇ u | p dx (cid:19) p . (2.2)Next, we define the notion of a weak solution of (1.1). Definition 2.5
A function u ∈ L ∞ ( R n ) is a weak subsolution of (1.1) if u ∈ W ,p loc (Ω) andfor every Ω ′ ⋐ Ω and nonnegative test functions φ ∈ W ,p (Ω ′ ) , we have ˆ Ω ′ |∇ u | p − ∇ u · ∇ φ dx + ˆ R n A ( u ( x, y ))( φ ( x ) − φ ( y )) dµ ≤ , (2.3) where A ( u ( x, y )) = | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) and dµ = K ( x, y ) dx dy. Analogously, a function u is a weak supersolution of (1.1) , if the integral in (2.3) is nonneg-ative for every nonnegative test functions φ ∈ W ,p (Ω ′ ) . A function u is a weak solution of (1.1) , if the equality holds in (2.3) for every φ ∈ W ,p (Ω ′ ) without a sign restriction. Remark 2.6
The boundedness assumption, together with Lemma 2.2 and Lemma 2.3, en-sures that Definition 2.5 is well stated and the tail defined in (3.5) is finite. Under theassumption u ∈ W ,p loc (Ω) ∩ W s,p ( R n ) , our main results Theorem 4.2, Theorem 5.1, Theorem8.2 and Theorem 8.3 hold true without the a priori boundedness assumption. In such a case,the local boundedness follows from Theorem 4.2. It follows directly from Defintion 2.5 that u is a weak subsolution of (1.1) if and only if − u is a weak supersolution of (1.1). Moreover, for any c ∈ R , u + c is a weak solution if and onlyif u is a weak solution. We discuss some further structural properties of weak solutions below.We denote the positive and negative parts of a ∈ R by a + = max { a, } and a − = max {− a, } ,respectively. Also, the barred integral sign denotes the corresponding integral average. Lemma 2.7
A function u is a weak solution of (1.1) if and only if u is a weak subsolutionand a weak supersolution of (1.1) .Proof. It follows immediately from Definition 2.5, that a weak solution u of (1.1) is aweak subsolution and a weak supersolution of (1.1). Conversely, assume that u is both weaksubsolution and supersolution of (1.1). Let Ω ′ ⋐ Ω and φ ∈ W ,p (Ω ′ ). Then φ + and φ − belong to W ,p (Ω ′ ). Since u is a weak subsolution, we have ˆ Ω ′ |∇ u | p − ∇ u · ∇ φ + dx + ˆ R n ˆ R n A ( u ( x, y ))( φ + ( x ) − φ + ( y )) dµ ≤ . (2.4)Analogously, since u is a weak supersolution, we have ˆ Ω ′ |∇ u | p − ∇ u · ∇ φ − dx + ˆ R n ˆ R n A ( u ( x, y ))( φ − ( x ) − φ − ( y )) dµ ≥ . (2.5) ixed local and nonlocal quasilinear equations φ = φ + − φ − , we obtain ˆ Ω ′ |∇ u | p − ∇ u · ∇ φ dx + ˆ R n ˆ R n A ( u ( x, y ))( φ ( x ) − φ ( y )) dµ ≤ . The reverse inequality holds by replacing φ with − φ . Hence, u is a weak solution of (1.1).Next, we show that the property of being a weak subsolution is preserved under takingthe positive part. Then, it follows immediately that u − is a subsolution of (1.1), whenever u is a supersolution of (1.1). Lemma 2.8
Assume that u is a weak subsolution of (1.1) . Then u + is a weak subsolutionof (1.1) .Proof. Consider functions u k = min { ku + , } , k = 1 , , . . . . Then ( u k ) k ∈ N is an increasingsequence of functions in W ,p loc (Ω) and 0 ≤ u k ≤ k ∈ N . Let φ ∈ C ∞ c (Ω ′ ) be anonnegative function. By choosing u k φ ∈ W ,p (Ω ′ ) as a test function in (2.3), we obtain0 ≥ ˆ Ω ′ |∇ u | p − ∇ u · ∇ ( u k φ ) dx + ˆ R n ˆ R n A ( u ( x, y ))( u k ( x ) φ ( x ) − u k ( y ) φ ( y )) dµ = I + I . (2.6) Estimate of I : We observe that I = ˆ Ω ′ |∇ u | p − ∇ u · ∇ ( u k φ ) dx = k ˆ Ω ′ ∩{ u ( y ).If u k ( x ) = 0, then u k ( y ) = 0. Hence, we have( u ( x ) − u ( y )) p − ( u k ( x ) φ ( x ) − u k ( y ) φ ( y )) = 0 . (2.8)If u k ( y ) >
0, then u ( y ) = u + ( y ). Under the assumption u ( x ) > u ( y ), it follows that u ( x ) = u + ( x ) and u k ( x ) > u k ( y ). This implies that( u ( x ) − u ( y )) p − ( u k ( x ) φ ( x ) − u k ( y ) φ ( y ))= ( u + ( x ) − u + ( y )) p − ( u k ( x ) φ ( x ) − u k ( y ) φ ( y )) ≥ ( u + ( x ) − u + ( y )) p − u k ( x )( φ ( x ) − φ ( y )) . (2.9)If u k ( y ) = 0 and u k ( x ) >
0, then u ( x ) > ≥ u ( y ) and hence( u ( x ) − u ( y )) p − ( u k ( x ) φ ( x ) − u k ( y ) φ ( y ))= ( u ( x ) − u ( y )) p − u k ( x ) φ ( x ) ≥ ( u + ( x ) − u + ( y )) p − u k ( x ) φ ( x ) ≥ ( u + ( x ) − u + ( y )) p − u k ( x )( φ ( x ) − φ ( y )) . (2.10) ixed local and nonlocal quasilinear equations A ( u ( x, y ))( u k ( x ) φ ( x ) − u k ( y ) φ ( y )) ≥ ( u + ( x ) − u + ( y )) p − u k ( x )( φ ( x ) − φ ( y )) . (2.11)When u ( x ) = u ( y ), the estimate (2.11) hods true. In case of u ( x ) < u ( y ), by interchangingthe roles of x and y in the above estimates, we arrive at A ( u ( x, y ))( u k ( x ) φ ( x ) − u k ( y ) φ ( y )) ≥ ( u + ( y ) − u + ( x )) p − u k ( y )( φ ( y ) − φ ( x )) . (2.12)Combining the estimates (2.7), (2.11) and (2.12) in (2.6) and letting k → ∞ , along with anapplication of the Lebesgue dominated convergence theorem, we obtain ˆ Ω ′ |∇ u + | p − ∇ u + · ∇ φ dx + ˆ R n ˆ R n A ( u + ( x, y ))( φ ( x ) − φ ( y )) dµ ≤ . (2.13)Then, by density for every φ ∈ W ,p (Ω ′ ), the estimate (2.13) holds. Hence, u + is a weaksubsolution of (1.1). The following energy estimate will be crucial for us.
Lemma 3.1
Let u be a weak subsolution of (1.1) and denote w = ( u − k ) + with k ∈ R .There exists a constant C = C ( p, Λ) such that ˆ B r ( x ) ψ p |∇ w | p dx + ˆ B r ( x ) ˆ B r ( x ) | w ( x ) ψ ( x ) − w ( y ) ψ ( y ) | p dµ ≤ C (cid:18) ˆ B r ( x ) w p |∇ ψ | p dx + ˆ B r ( x ) ˆ B r ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ + ess sup y ∈ supp ψ ˆ R n \ B r ( x ) w ( y ) p − | x − y | n + ps dy · ˆ B r ( x ) wψ p dx (cid:19) , (3.1) whenever B r ( x ) ⊂ Ω and ψ ∈ C ∞ c ( B r ) is a nonnegative function. If u is a weak supersolutionof (1.1) , the estimate in (3.1) holds with w = ( u − k ) − .Proof. Let u be a weak subsolution of (1.1). For w = ( u − k ) + , by choosing φ = wψ p asa test function in (2.3), we obtain0 ≥ ˆ B r ( x ) |∇ u | p − ∇ u · ∇ ( wψ p ) dx + ˆ R n ˆ R n A ( u ( x, y ))( w ( x ) ψ ( x ) p − w ( y ) ψ ( y ) p ) dµ = I + J. (3.2)Proceeding as in the proof of [6, Proposition 3.1], for some constants c = c ( p ) > C = C ( p ) >
0, we have I = ˆ B r ( x ) |∇ u | p − ∇ u · ∇ ( wψ p ) dx ≥ c ˆ B r ( x ) ψ p |∇ w | p dx − C ˆ B r ( x ) w p |∇ ψ | p dx. (3.3) ixed local and nonlocal quasilinear equations c = c ( p, Λ) > C = C ( p, Λ) >
0, we have J = ˆ R n ˆ R n A ( u ( x, y ))( w ( x ) ψ ( x ) p − w ( y ) ψ ( y ) p ) dµ ≥ c ˆ B r ( x ) ˆ B r ( x ) | w ( x ) ψ ( x ) − w ( y ) ψ ( y ) | p dµ − C ˆ B r ( x ) ˆ B r ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ − C ess sup y ∈ supp φ ˆ R n \ B r ( x ) w ( y ) p − | x − y | n + ps dy · ˆ B r ( x ) wψ p dx. (3.4)By applying (3.3) and (3.4) in (3.2), we obtain (3.1). In the case of a weak supersolution, theestimate in (3.1) follows by applying the obtained result for − u .Next we define the tail which appears in estimates throughout the article. Definition 3.2
The tail of a weak subsolution or weak supersolution u of (1.1) (Definition2.5) with respect to a ball B r ( x ) is defined by Tail( u ; x , r ) = (cid:18) r p ˆ R n \ B r ( x ) | u ( y ) | p − | y − x | n + ps dy (cid:19) p − . (3.5)We prove an energy estimate which will be crucial to obtain a reverse H¨older inequalityfor weak supersolutions of (1.1). Lemma 3.3
Let q ∈ (1 , p ) and d > . Assume that u is a weak supersolution of (1.1) suchthat u ≥ in B R ( x ) ⊂ Ω and denote by w = ( u + d ) p − qp . There exists a constant c = c ( p, Λ) such that ˆ B r ( x ) ψ p |∇ w | p dx ≤ c (cid:18) ( p − q ) p ( q − pp − ˆ B r ( x ) w p |∇ ψ | p dx + ( p − q ) p ( q − p ˆ B r ( x ) ˆ B r ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ + ( p − q ) p ( q − (cid:18) ess sup z ∈ supp ψ ˆ R n \ B r ( x ) K ( z, y ) dy + d − p R − p Tail( u − ; x , R ) p − (cid:19) ˆ B r ( x ) w p ψ p dx (cid:19) , (3.6) whenever B r ( x ) ⊂ B R ( x ) and ψ ∈ C ∞ c ( B r ( x )) is a nonnegative function. Here Tail( · ) isdefined in (3.5) .Proof. Let d > v = u + d and q ∈ [1 + ǫ, p − ǫ ] for ǫ > v is a ixed local and nonlocal quasilinear equations φ = v − q ψ p as a test function in (2.3), we obtain0 ≤ ˆ B r ( x ) |∇ v | p − ∇ v · ∇ ( v − q ψ p ) dx + ˆ B r ( x ) ˆ B r ( x ) A ( v ( x, y ))( v ( x ) − q ψ ( x ) p − v ( y ) − q ψ ( y ) p ) dµ + 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( v ( x, y )) v ( x ) − q ψ ( x ) p dµ = I + I + 2 I . (3.7) Estimate of I : We observe that I = ˆ B r ( x ) |∇ v | p − ∇ v · ∇ ( v − q ψ p ) dx ≤ (1 − q ) ˆ B r ( x ) v − q |∇ v | p ψ p dx + p ˆ B r ( x ) v − q |∇ ψ ||∇ v | p − ψ p − dx = (1 − q ) J + J , (3.8)where J = ˆ B r ( x ) v − q |∇ v | p ψ p dx and J = p ˆ B r ( x ) v − q |∇ ψ ||∇ v | p − ψ p − dx. Estimate of J : By Young’s inequality, we obtain J = p ˆ B r ( x ) v − q |∇ ψ ||∇ v | p − ψ p − dx ≤ q − J + c ( p )( q − p − ˆ B r ( x ) |∇ ψ | p v p − q dx. (3.9)By applying (3.9) in (3.8), we have I ≤ − q ˆ B r ( x ) v − q |∇ v | p ψ p dx + c ( p )( q − p − ˆ B r ( x ) |∇ ψ | p v p − q dx = − q − (cid:16) pp − q (cid:17) p ˆ B r ( x ) (cid:12)(cid:12) ∇ ( v p − qp ) (cid:12)(cid:12) p ψ p dx + c ( p )( q − p − ˆ B r ( x ) |∇ ψ | p v p − q dx. (3.10) Estimate of I and I : Following the proof of [13, Lemma 5.1] for w = v p − qp , we obtain I = ˆ B r ( x ) ˆ B r ( x ) A ( v ( x, y ))( v ( x ) − q ψ ( x ) p − v ( y ) − q ψ ( y ) p ) dµ ≤ − c ( p, q ) ˆ B r ( x ) ˆ B r ( x ) | w ( x ) − w ( y ) | p ψ ( y ) p dµ + c ( p )( q − p − ˆ B r ( x ) ˆ B r ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ (3.11) ixed local and nonlocal quasilinear equations I = 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( v ( x, y )) v ( x ) − q ψ ( x ) p dµ ≤ c ( p, Λ) (cid:18) ess sup z ∈ supp ψ ˆ R n \ B r ( x ) K ( z, y ) dy + d − p ˆ R n \ B r ( x ) u ( y ) p − − | y − x | − n − sp dy (cid:19) ˆ B r ( x ) w p ψ p dx. (3.12)By applying (3.10), (3.11) and (3.12) in (3.7), we obtain (3.6).Next, we obtain a logarithmic energy estimate. Lemma 3.4
Assume that u is a weak supersolution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω .There exists a constant c = c ( n, p, s, Λ) such that ˆ B r ( x ) |∇ log( u + d ) | p dx + ˆ B r ( x ) ˆ B r ( x ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ≤ cr n (cid:0) r − p + r − sp + d − p R − p Tail( u − ; x , R ) p − (cid:1) , (3.13) whenever B r ( x ) ⊂ B R ( x ) and d > . Here Tail( · ) is given by (3.5) .Proof. Let ψ ∈ C ∞ c ( B r ( x )) be such that 0 ≤ ψ ≤ B r ( x ), ψ = 1 in B r ( x ), and |∇ ψ | ≤ r in B r ( x ). By choosing φ = ( u + d ) − p ψ p as a test function in (2.3), we obtain0 ≤ ˆ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))(( u ( x ) + d ) − p ψ ( x ) p − ( u ( y ) + d ) − p ψ ( y ) p ) dµ + 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))( u ( x ) + d ) − p ψ ( x ) p dµ + ˆ B r ( x ) |∇ u | p − ∇ u · ∇ (( u + d ) − p ψ p ) dx = I + I + I . (3.14) Estimate of I and I : Following the proof [14, Lemma 1.3] and using the properties of ψ ,we obtain I = ˆ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))(( u ( x ) + d ) − p ψ ( x ) p − ( u ( y ) + d ) − p ψ ( y ) p ) dµ ≤ − c ˆ B r ( x ) ˆ B r ( x ) K ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ψ ( y ) p dx dy + cr n − sp (3.15)and I = 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))( u ( x ) + d ) − p ψ ( x ) p dµ ≤ cd − p r n R − p Tail( u − ; x , R ) p − + cr n − sp , (3.16) ixed local and nonlocal quasilinear equations c = c ( n, p, s, Λ).
Estimate of I : Arguing similarly as in the proof of [25, Lemma 3.4], we have I = ˆ B r ( x ) |∇ u | p − ∇ u · ∇ (( u + d ) − p ψ p ) dx ≤ − c ˆ B r ( x ) |∇ log( u + d ) | p dx + cr n − p , (3.17)for some positive constant c = c ( p ).Hence using (3.15), (3.16) and (3.17) in (3.14) along with the fact that ψ ≡ B r ( x ), theestimate (3.13) follows.As a consequence of Lemma 3.4, we have the following result. Corollary 3.5
Assume that u is a weak solution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω .Let a, d > , b > and denote v = min (cid:26)(cid:18) log (cid:18) a + du + d (cid:19)(cid:19) + , log b (cid:27) . There exists a constant c = c ( n, p, s, Λ) such that B r ( x ) | v − v B r ( x ) | p dx ≤ c (cid:16) d − p (cid:16) rR (cid:17) p Tail( u − ; x , R ) p − (cid:17) , (3.18) whenever B r ( x ) ⊂ B R ( x ) with r ∈ (0 , . Here Tail( · ) is given by (3.5) .Proof. By the Poincar´e inequality from [20, Theorem 2], we have B r ( x ) | v − v B r ( x ) | p dx ≤ cr p − n ˆ B r ( x ) |∇ v | p dx, (3.19)for a constant c = c ( n, p ). Now since v is a truncation of the sum of a constant and log( u + d ),we have ˆ B r ( x ) |∇ v | p dx ≤ ˆ B r ( x ) |∇ log( u + d ) | p dx. (3.20)The estimate in (3.18) follows by employing (3.13) in (3.20) along with (3.19) and the fact r ∈ (0 , We need the following real analysis lemma. For the proof of Lemma 4.1, see [16, Lemma 4.1].
Lemma 4.1
Let ( Y j ) ∞ j =0 be a sequence of positive real numbers such that Y ≤ c − β b − β and Y j +1 ≤ c b j Y βj , j = 0 , , , . . . for some constants c , b > and β > . Then, we have lim j →∞ Y j = 0 . ixed local and nonlocal quasilinear equations Theorem 4.2 ( Local boundedness ). Let u be a weak subsolution of (1.1) . There exists apositive constant c = c ( n, p, s, Λ) , such that ess sup B r ( x ) u ≤ cδ Tail (cid:0) u + ; x , r (cid:1) + cδ − ( p − κp ( κ − (cid:18) B r ( x ) u p + dx (cid:19) p , (4.1) whenever B r ( x ) ⊂ Ω with r ∈ (0 , and δ ∈ (0 , . Here κ and Tail( · ) are given by (2.1) and (3.5) , respectively.Proof. Let B r ( x ) ⊂ Ω with r ∈ (0 , j = 0 , , , . . . , we denote r j = r − j ) , ¯ r j = r j + r j +1 ,B j = B r j ( x ) and ¯ B j = B ¯ r j ( x ). Let ψ j ∈ C ∞ c ( ¯ B j ) be a cutoff function such that 0 ≤ ψ j ≤ B j , ψ j = 1 in B j +1 and |∇ ψ j | ≤ j +3 r . For j = 0 , , , . . . and k, ¯ k ∈ R , we denote k j = k + (1 − − j )¯ k, ¯ k j = k j + k j +1 ,w j = ( u − k j ) + and ¯ w j = ( u − ¯ k j ) + . Then (cid:16) ¯ k j +2 (cid:17) p ( κ − κ (cid:18) B j +1 w pj +1 dx (cid:19) κ = ( k j +1 − ¯ k j ) p ( κ − κ (cid:18) B j +1 w pj +1 dx (cid:19) κ ≤ (cid:18) ¯ B j | ¯ w j ψ j | pκ dx (cid:19) κ , (4.2)where κ is given by (2.1). By the Sobolev inequality in (2.2), we obtain (cid:18) ¯ B j | ¯ w j ψ j | pκ dx (cid:19) κ ≤ cr p − n ˆ B j |∇ ( ¯ w j ψ j ) | p dx ≤ cr p − n (cid:18) ˆ B j ¯ w pj |∇ ψ j | p dx + ˆ B j ψ pj |∇ ¯ w j | p dx (cid:19) = I + I , (4.3)for a constant c = c ( n, p, s ). Estimate of I : Using the properties of ψ j , we have I = c ( n, p, s ) r p − n ˆ B j ¯ w pj |∇ ψ j | p dx ≤ c ( n, p, s )2 jp B j w pj dx. (4.4) ixed local and nonlocal quasilinear equations Estimate of I : By Lemma 3.1, we obtain I = c ( n, p, s ) r p − n ˆ B j ψ pj |∇ ¯ w j | p dx ≤ C ( n, p, s, Λ) r p − n (cid:18) ˆ B j ¯ w pj |∇ ψ j | p dx + ˆ B j ˆ B j max { ¯ w j ( x ) , ¯ w j ( y ) } p | ψ j ( x ) − ψ j ( y ) | p dµ + ˆ B j ¯ w j ( y ) ψ j ( y ) p dy · ess sup y ∈ supp ψ j ˆ R n \ B j ¯ w j ( y ) p − K ( x, y ) dx (cid:19) = J + J + J . (4.5) Estimate of J and J : Proceeding similarly as in the proof of [14, Theorem 1.1] and againusing the properties of ψ j , we obtain J i ≤ c ( n, p, s, Λ)2 jp B j w pj dx, j = 1 , , (4.6)for every r ∈ (0 , Estimate of J : We observe that w pj ≥ (¯ k j − k j ) p − ¯ w j . For any δ ∈ (0 , J = ˆ B j ¯ w j ( y ) ψ j ( y ) p dy · ess sup y ∈ supp ψ j ˆ R n \ B j ¯ w j ( y ) p − K ( x, y ) dx = c j ( n + ps ) r p B j ¯ w j ( y ) p (¯ k j − k j ) p − dy ˆ R n \ B j w j ( x ) p − | x − x | n + ps dx ≤ c j ( n + ps + p − ¯ k p − Tail( w ; x , r ) p − B j w j ( y ) p dy ≤ c j ( n + ps + p − δ − p B j w j ( y ) p dy, (4.7)whenever ¯ k ≥ δ Tail( w ; x , r ). Here we used the fact that | x − x || x − y | ≤ | x − y | + | y − x || x − y | ≤ r j r j − ¯ r j ≤ j +4 , which holds for x ∈ R n \ B j and y ∈ supp ψ j = ¯ B j .By applying (4.6) and (4.7) in (4.5), we obtain I ≤ c ( n, p, s, Λ)2 j ( n + ps + p − δ − p B j w pj dx (4.8)for every δ ∈ (0 , (cid:18) B j | ¯ w j ψ j | pκ dx (cid:19) κ ≤ c ( n, p, s, Λ)2 j ( n + ps + p − δ − p B j w pj dx. (4.9) ixed local and nonlocal quasilinear equations Y j = (cid:18) B j w pj dx (cid:19) p , and ¯ k = δ Tail( w ; x , r ) + c β b β (cid:18) B r u p + dx (cid:19) p , where c = c ( n, p, s, Λ) δ (1 − p ) κp , b = 2 ( n + ps + p − p + κ − κ ) κ , β = κ − . From (4.2) and (4.9) we obtain Y j +1 ¯ k ≤ c ( n, p, s, Λ)2 j ( n + ps + p − p + κ − κ ) κ δ (1 − p ) κp (cid:16) Y j ¯ k (cid:17) κ . (4.10)Moreover, by the definition of ¯ k above we have Y ¯ k ≤ c − β b − β . From Lemma 4.1, we obtain Y j → j → ∞ . This implies thatess sup B r ( x ) u ≤ k + ¯ k, which gives (4.1) by choosing k = 0. The following local H¨older continuity result for weak solutions of (1.1) follows from Lemma5.2 below.
Theorem 5.1 ( H¨older continuity ). Let u be a weak solution of (1.1) . Then u is locallyH¨older continuous in Ω . Moreover, there exist constants α ∈ (0 , pp − ) and c = c ( n, p, s, Λ) ,such that osc B ρ ( x ) u = ess sup B ρ ( x ) u − ess inf B ρ ( x ) u ≤ c (cid:16) ρr (cid:17) α (cid:18) Tail( u ; x , r ) + (cid:18) B r ( x ) | u | p dx (cid:19) p (cid:19) , (5.1) whenever B r ( x ) ⊂ Ω with r ∈ (0 , and ρ ∈ (0 , r ] . Here Tail( · ) is given by (3.5) . Lemma 5.2
Let u be a weak solution of (1.1) and B r ( x ) ⊂ B R ( x ) ⊂ Ω with r ∈ (0 , .For η ∈ (0 , ] , we set r j = η j r for j = 0 , , , . . . . Denote ω ( r ) = Tail (cid:0) u ; x , r (cid:1) + c (cid:18) ˆ B r ( x ) | u | p dx (cid:19) p , ixed local and nonlocal quasilinear equations where Tail( · ) is given by (3.5) , c = c ( n, p, s, Λ) is the constant in (4.1) and let ω ( r j ) = (cid:16) r j r (cid:17) α ω ( r ) , j = 1 , , . . . , for some α ∈ (0 , pp − ) . Then osc B rj ( x ) u ≤ ω ( r j ) , j = 0 , , , . . . . (5.2) Proof.
By Lemma 2.7 and Lemma 2.8, the functions u + and ( − u ) + are weak subsolutionsof (1.1). Then applying Theorem 4.2 with u + and ( − u ) + , it is easy to observe that (5.2) holdstrue for j = 0. The estimate in (5.2) follows from Lemma 3.1, Corollary 3.5, the Sobolevinequality along with the fact that r ∈ (0 ,
1] and repeating the induction argument as in theproof of [14, Lemma 5.1].
The following tail estimate will be useful for us.
Lemma 6.1
Let u be a weak solution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω . There existsa constant c = c ( n, p, s, Λ) such that Tail( u + ; x , r ) ≤ c ess sup B r ( x ) u + c (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (6.1) whenever < r < R with r ∈ (0 , . Here Tail( · ) is given by (3.5) .Proof. Let M = ess sup B r ( x ) u and ψ ∈ C ∞ c ( B r ( x )) be a cutoff function such that0 ≤ ψ ≤ B r ( x ), ψ = 1 in B r ( x ) and |∇ ψ | ≤ r in B r ( x ). By letting w = u − M andchoosing φ = wψ p as a test function in (2.3) we obtain0 = ˆ B r ( x ) |∇ u | p − ∇ u · ∇ ( wψ p ) dx + ˆ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))( w ( x ) ψ ( x ) p − w ( y ) ψ ( y ) p ) dµ + 2 ˆ B r ( x ) ˆ R n \ B r ( x ) A ( u ( x, y )) w ( x ) ψ ( x ) p dµ = I + I + I . (6.2) Estimate of I : By Young’s inequality, the estimate |∇ w | p − ∇ w · ∇ ( wψ p )( x ) = |∇ w | p ψ p + pψ p − w |∇ w | p − ∇ w · ∇ ψ ≥ |∇ w | p ψ p − c ( p ) | w | p |∇ ψ | p ≥ − c ( p ) M p |∇ ψ | p , ixed local and nonlocal quasilinear equations B r ( x ). By the properties of ψ , we have I = ˆ B r ( x ) |∇ u | p − ∇ u · ∇ ( wψ p ) dx ≥ − c ( p ) M p r − p | B r ( x ) | . (6.3) Estimate of I and I : Proceeding along the lines of the proof of [13, Lemma 4.2] and usingthe fact that r ∈ (0 , c = c ( n, p, s, Λ) > I = ˆ B r ( x ) ˆ B r ( x ) A ( u ( x, y ))( w ( x ) ψ ( x ) p − w ( y ) ψ ( y ) p ) dµ ≥ − cM p r − p | B r ( x ) | , (6.4)and I = 2 ˆ B r ( x ) ˆ R n \ B r ( x ) A ( u ( x, y )) w ( x ) ψ ( x ) p dµ ≥ cM r − p Tail( u + ; x , r ) p − | B r ( x ) | − cM R − p Tail( u − ; x , R ) p − | B r ( x ) |− cM p r − p | B r ( x ) | . (6.5)The estimate in (6.1) follows by applying (6.3), (6.4) and (6.5) in (6.2). The following lemma shows that the expansion of positivity technique applies to mixed prob-lems.
Lemma 7.1
Let u be a weak supersolution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω . Assume k ≥ and there exists τ ∈ (0 , such that (cid:12)(cid:12) B r ( x ) ∩ { u ≥ k } (cid:12)(cid:12) ≥ τ | B r ( x ) | , (7.1) for some r ∈ (0 , with < r < R . There exists a constant δ = δ ( n, p, s, Λ , τ ) ∈ (0 , ) suchthat ess inf B r ( x ) u ≥ δk − (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (7.2) where Tail( · ) is given by (3.5) .Proof. We prove the lemma in two steps.
Step 1.
Under the assumption in (7.1), we claim that there exists a constant c = c ( n, p, s, Λ)such that (cid:12)(cid:12)(cid:12) B r ( x ) ∩ n u ≤ δk − (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) − ǫ o(cid:12)(cid:12)(cid:12) ≤ c τ log δ | B r ( x ) | (7.3)for every δ ∈ (0 , ) and for every ǫ > ixed local and nonlocal quasilinear equations ǫ > ψ ∈ C ∞ c ( B r ( x )) be a cutoff function such that 0 ≤ ψ ≤ B r ( x ), ψ = 1 in B r ( x ) and |∇ ψ | ≤ r in B r ( x ). We denote w = u + t ǫ , where t ǫ = 12 (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) + ǫ. Since w is a weak supersolution of (1.1), we can choose φ = w − p ψ p as a test function in (2.3)to obtain 0 ≤ ˆ B r ( x ) |∇ w | p − ∇ w · ∇ ( w − p ψ p ) dx + ˆ B r ( x ) ˆ B r ( x ) A ( w ( x, y ))( w ( x ) − p ψ ( x ) p − w ( y ) − p ψ ( y ) p ) dµ + 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( w ( x, y )) w ( x ) − p ψ ( x ) p dµ = I + I + I . (7.4) Estimate of I : Proceeding similarly as in the proof of [25, Lemma 3.4], we obtain a constant c = c ( p ) > I = ˆ B r ( x ) |∇ w | p − ∇ w · ∇ ( w − p ψ p ) dx ≤ − c ˆ B r ( x ) |∇ log w | p dx + cr n − p . (7.5) Estimate of I and I : Arguing as in the proof of [13, Lemma 3.1] and using the fact that r ∈ (0 , c = c ( n, p, s, Λ) > I = ˆ B r ( x ) ˆ B r ( x ) A ( w ( x, y ))( w ( x ) − p ψ ( x ) p − w ( y ) − p ψ ( y ) p ) dµ ≤ − c ˆ B r ( x ) ˆ B r ( x ) (cid:12)(cid:12)(cid:12)(cid:12) log w ( x ) w ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p dµ + cr n − p (7.6)and I = 2 ˆ R n \ B r ( x ) ˆ B r ( x ) A ( w ( x, y )) w ( x ) − p ψ ( x ) p dµ ≤ cr n − p . (7.7)By (7.5), (7.6) and (7.7) in (7.4), we obtain ˆ B r ( x ) |∇ log w | p dx + ˆ B r ( x ) ˆ B r ( x ) (cid:12)(cid:12)(cid:12)(cid:12) log w ( x ) w ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p dµ ≤ cr n − p . (7.8)For δ ∈ (0 , ), we define the function v = (cid:18) min (cid:26) log 12 δ , log k + t ǫ w (cid:27)(cid:19) + . ixed local and nonlocal quasilinear equations ˆ B r ( x ) |∇ v | p dx ≤ ˆ B r ( x ) |∇ log w | p dx ≤ cr n − p . (7.9)From (7.9) along with H¨older’s inequality and Poincar´e inequality (see [20, Theorem 2]), weobtain ˆ B r ( x ) | v − v B r ( x ) | dx ≤ cr np ′ (cid:18) ˆ B r ( x ) |∇ v | p dx (cid:19) p ≤ c | B r ( x ) | , (7.10)where p ′ = pp − . We observe that { v = 0 } = { w ≥ k + t ǫ } = { u ≥ k } . By the assumption(7.1), it follows that (cid:12)(cid:12) B r ( x ) ∩ { v = 0 } (cid:12)(cid:12) ≥ τ n | B r ( x ) | . (7.11)Following the proof of [13, Lemma 3.1] and using (7.11), we obtainlog 12 δ = 1 | B r ( x ) ∩ { v = 0 }| ˆ B r ( x ) ∩{ v =0 } (cid:16) log 12 δ − v ( x ) (cid:17) dx ≤ n τ (cid:16) log 12 δ − v B r (cid:17) . (7.12)Now integrating (7.12) over the set B r ( x ) ∩ { v = log δ } and using (7.10), we get for someconstant c = c ( n, p, s, Λ) > (cid:12)(cid:12)(cid:12)n v = log 12 δ o ∩ B r ( x ) (cid:12)(cid:12)(cid:12) log 12 δ ≤ n τ ˆ B r ( x ) | v − v B r ( x ) | dx ≤ c τ | B r ( x ) | . Hence, for any δ ∈ (0 , ), we have (cid:12)(cid:12) B r ( x ) ∩ { w ≤ δ ( k + t ǫ ) } (cid:12)(cid:12) ≤ c τ δ | B r ( x ) | . This implies (7.3).
Step 2.
We claim that, for every ǫ >
0, there exists a constant δ = δ ( n, p, s, Λ , τ ) ∈ (0 , )such that ess inf B r ( x ) u ≥ δk − (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) − ǫ. (7.13)As a consequence of (7.13), the property (7.2) follows.To prove (7.13), without loss of generality, we may assume that δk ≥ (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) + 2 ǫ. (7.14)Otherwise (7.13) holds tue, since u ≥ B R ( x ).Let ρ ∈ [ r, r ] and ψ ∈ C ∞ c ( B ρ ( x )) be a cutoff function such that 0 ≤ ψ ≤ B ρ ( x ).For any l ∈ ( δk, δk ), from Lemma 3.1 and the proof of [13, Lemma 3.2] for w = ( l − u ) + , we ixed local and nonlocal quasilinear equations ˆ B ρ ( x ) ψ p |∇ w | p dx + ˆ B ρ ( x ) ˆ B ρ ( x ) | w ( x ) ψ ( x ) − w ( y ) ψ ( y ) | p dµ ≤ ˆ B ρ ( x ) w p |∇ ψ | p dx + ˆ B ρ ( x ) ˆ B ρ ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ + cl ess sup x ∈ supp ψ ˆ R n \ B ρ ( x ) (cid:0) l + u ( y ) − (cid:1) p − K ( x, y ) dy · (cid:12)(cid:12) B ρ ( x ) ∩ { u < l } (cid:12)(cid:12) = J + J + J . (7.15)We apply Lemma 4.1 to conclude the proof. For j = 0 , , , . . . , we denote l = k j = δk + 2 − j − δk, ρ = ρ j = 4 r + 2 − j r, ˆ ρ j = ρ j + ρ j +1 . (7.16)Then l ∈ ( δk, δk ), ρ , ˆ ρ j ∈ (4 r, r ) and k j − k j +1 = 2 − j − δk ≥ − j − k j for every j = 0 , , , . . . . Set B j = B ρ j ( x ) , ˆ B j = B ˆ ρ j and we observe that w j = ( k j − u ) + ≥ − j − k j χ { u
0. We choose c = c , b = 2 ( n +2 p + ps ) κ > β = κ − > k = 32 δk ≤ δk − (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) − ǫ. By (7.3) we have Y ≤ (cid:12)(cid:12) B r ( x ) ∩ (cid:8) u ≤ δk − (cid:0) rR (cid:1) pp − Tail( u − ; x , R ) − ǫ (cid:9)(cid:12)(cid:12) | B r ( x ) | ≤ c τ log δ (7.23)for some constant c = c ( n, p, s, Λ) > δ ∈ (0 , ). Using (7.23) we choose δ = δ ( n, p, s, Λ , τ ) ∈ (0 , ) such that0 < δ = 14 exp (cid:18) − c c β b β τ (cid:19) < , so that the estimate Y ≤ c − β b − β holds. By Lemma 4.1 we conclude that Y j → j → ∞ .Therefore, we have ess inf B r ( x ) u ≥ δk, which gives (7.13) and so (7.2) holds. ixed local and nonlocal quasilinear equations As the proof of [13, Lemma 4.1], along with an application of Lemma 7.1, we obtain thefollowing preliminary version of the weak Harnack inequality, compared to Theorem 8.3.
Lemma 8.1
Let u be a weak supersolution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω . Thereexist constants η = η ( n, p, s, Λ) ∈ (0 , and c = c ( n, p, s, Λ) ≥ such that (cid:18) B r ( x ) u η dx (cid:19) η ≤ c ess inf B r ( x ) u + c (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (8.1) whenever B r ( x ) ⊂ B R ( x ) with r ∈ (0 , . Here Tail( · ) is defined in (3.5) . By applying Lemma 2.7, Theorem 4.2, Lemma 6.1 and Lemma 8.1, the following Harnackinequality follows with a similar argument as in the proof of [13, Theorem 1.1].
Theorem 8.2 ( Harnack inequality ). Let u be a weak solution of (1.1) such that u ≥ in B R ( x ) ⊂ Ω . There exists a constant c = c ( n, p, s, Λ) such that ess sup B r ( x ) u ≤ c ess inf B r ( x ) u + c (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (8.2) whenever B r ( x ) ⊂ B R ( x ) and r ∈ (0 , . Here Tail( · ) is given by (3.5) . We have the following weak Harnack inequality for supersolutions of (1.1).
Theorem 8.3 ( Weak Harnack inequality ). Let u be a weak supersolution of the problem (1.1) such that u ≥ in B R ( x ) ⊂ Ω . There exists a constant c = c ( n, p, s, Λ) such that (cid:18) B r ( x ) u l dx (cid:19) l ≤ c ess inf B r ( x ) u + c (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (8.3) whenever B r ( x ) ⊂ B R ( x ) , r ∈ (0 , and < l < κ ( p − . Here κ and Tail( · ) are given by (2.1) and (3.5) , respectively.Proof. We prove the result for 1 < p < n . For p ≥ n , the result follows in a similarway. Let r ∈ (0 , < τ ′ < τ ≤ and we choose ψ ∈ C ∞ c ( B τr ( x )) such that 0 ≤ ψ ≤ B τr ( x ), ψ = 1 in B τ ′ r ( x ) and |∇ ψ | ≤ τ − τ ′ ) r . For d >
0, we set v = u + d and w = ( u + d ) p − qp . Proceeding as in the proof of [13, Theorem 1.2], there exists a constant c ( p ) such that I = ˆ B r ( x ) w p |∇ ψ | p dx ≤ c ( p ) r − p ( τ − τ ′ ) p ˆ B τr ( x ) w p dx, (8.4) ixed local and nonlocal quasilinear equations I = ˆ B r ( x ) ˆ B r ( x ) max { w ( x ) , w ( y ) } p | ψ ( x ) − ψ ( y ) | p dµ ≤ c ( p ) r − p ( τ − τ ′ ) p ˆ B τr ( x ) w p dx. (8.5)Assume that Tail( u − ; x , R ) is positive. Then for any ǫ > r ∈ (0 ,
1] choosing d = 12 (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) + ǫ > , and noting that ess sup z ∈ supp ψ ˆ R n \ B r ( x ) K ( z, y ) dy ≤ c ( p ) r − p , (8.6)we obtain I = (cid:18) ess sup z ∈ supp ψ ˆ R n \ B r ( x ) K ( z, y ) dy + d − p R − p Tail( u − ; x , R ) p − (cid:19) ˆ B r ( x ) w p ψ p dx ≤ c ( p ) r − p ( τ − τ ′ ) p ˆ B τr ( x ) w p dx. (8.7)If Tail( u − ; x , R ) = 0, we can choose an arbitrary d = ǫ > ψ ≡ B τ ′ r , r ∈ (0 , p ∗ = npn − p with 1 < p < n , (cid:18) B τ ′ r ( x ) v n ( p − q ) n − p dx (cid:19) pp ∗ = (cid:18) B τ ′ r ( x ) w p ∗ dx (cid:19) pp ∗ ≤ (cid:18) B τr ( x ) | wψ | p ∗ dx (cid:19) pp ∗ ≤ ( τ r ) p − n ˆ B τr ( x ) |∇ ( wψ ) | p dx ≤ c ( τ − τ ′ ) p B τr ( x ) w p dx, (8.8)where c = c ( n, p, s, Λ). Using q ∈ (1 , p ) and the Moser iteration technique as in [23, Theorem8.18] and [30, Theorem 1.2], we get (cid:18) B r ( x ) v l dx (cid:19) l ≤ c (cid:18) B r ( x ) v l ′ dx (cid:19) l ′ , < l ′ < l < n ( p − n − p . (8.9)Let η ∈ (0 ,
1) be given by Lemma 8.1 and then choosing l ′ = η ∈ (0 ,
1) and observing that (cid:18) B r ( x ) u l dx (cid:19) l ≤ (cid:18) B r ( x ) v l dx (cid:19) l , we obtain from (8.9) (cid:18) B r ( x ) u l dx (cid:19) l ≤ ess inf B r ( x ) v + c ( n, p, s, Λ) (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) , (8.10)for all 0 < l ′ < l < n ( p − n − p . Now for any ǫ >
0, choosing d = 12 (cid:16) rR (cid:17) pp − Tail( u − ; x , R ) + ǫ, in (8.10) and then, letting ǫ →
0, the result follows. ixed local and nonlocal quasilinear equations Before stating our results on pointwise behavior, we discuss a result from Liao [27]. Let u bea measurable function that is locally essentially bounded below in Ω. Let ρ ∈ (0 ,
1] be suchthat B ρ ( y ) ⊂ Ω. Assume that a, c ∈ (0 , M > µ − ≤ ess inf B ρ ( y ) u . Following [27],we say that u satisfies the property ( D ), if there exists a constant τ ∈ (0 ,
1) depending on a, c, M, µ − and other data (may depend on the partial differential equation and will be madeprecise in Lemma 9.4), but independent of ρ , such that (cid:12)(cid:12) { u ≤ µ − + M } ∩ B ρ ( y ) (cid:12)(cid:12) ≤ τ | B ρ ( y ) | , implies that u ≥ µ − + aM almost everywhere in B cρ ( y ).Moreover, for u ∈ L (Ω), we define the set of Lebesgue points of u by F = (cid:26) x ∈ Ω : | u ( x ) | < ∞ , lim r → B r ( x ) | u ( x ) − u ( y ) | dy = 0 (cid:27) . Note that, by the Lebesgue differentiation theorem, |F | = | Ω | .The following result follows from [27, Theorem 2.1]. Lemma 9.1
Let u be a measurable function that is locally integrable and locally essentiallybounded below in Ω . Assume that u satisfies the property ( D ) . Then u ( x ) = u ∗ ( x ) for every x ∈ F , where u ∗ ( x ) = lim r → ess inf y ∈ B r ( x ) u ( y ) . In particular, u ∗ is a lower semicontinuous representative of u in Ω . Since u is assumed to be locally essentially bounded below, the lower semicontinuousregularization u ∗ ( x ) is well defined at every point x ∈ Ω. Our final regularity results statedare consequences of Lemma 9.1 and Lemma 9.4 below.
Theorem 9.2 ( Lower semicontinuity ). Let u be a weak supersolution of (1.1) . Then u ( x ) = u ∗ ( x ) = lim r → ess inf y ∈ B r ( x ) u ( y ) for every x ∈ F . In particular, u ∗ is a lower semicontinuous representative of u in Ω . As a Corollary Theorem 9.2, we have the following result.
Corollary 9.3 ( Upper semicontinuity ). Let u be a weak subsolution of (1.1) . Then u ( x ) = u ∗ ( x ) = lim r → ess sup y ∈ B r ( x ) u ( y ) for every x ∈ F . In particular, u ∗ is an upper semicontinuous representative of u in Ω . We prove a De Giorgi type lemma for weak supersolutions of (1.1). ixed local and nonlocal quasilinear equations Lemma 9.4
Let u be a weak supersolution of (1.1) . Let M > , a ∈ (0 , , B r ( x ) ⊂ Ω with r ∈ (0 , and µ − ≤ ess inf R n u . There exists a constant τ = τ ( n, p, s, Λ , a, M, µ − ) ∈ (0 , such that if (cid:12)(cid:12) { u ≤ µ − + M } ∩ B r ( x ) (cid:12)(cid:12) ≤ τ | B r ( x ) | , then u ≥ µ − + aM almost everywhere in B r ( x ) .Proof. Without loss of generality, we may assume that x = 0. For j = 0 , , , . . . , wedenote k j = µ − + aM + (1 − a ) M j , ¯ k j = k j + k j +1 ,r j = 3 r r j +2 , ¯ r j = r j + r j +1 , (9.1) B j = B r j (0), ¯ B j = B ¯ r j (0), w j = ( k j − u ) + and ¯ w j = (¯ k j − u ) + . We observe that B j +1 ⊂ ¯ B j ⊂ B j , ¯ k j < k j and hence ¯ w j ≤ w j . Let ψ j ∈ C ∞ c ( B j ) be a cutoff function satisfying 0 ≤ ψ j ≤ B j , ψ j = 1 in B j +1 , |∇ ψ j | ≤ j +3 r and dist(supp ψ j , R n \ B j ) ≥ − j − r .By applying Lemma 3.1 to w j , we obtain ˆ B j ˆ B j | w j ( x ) ψ j ( x ) − w j ( y ) ψ j ( y ) | p dµ + ˆ B j |∇ w j | p ψ pj dx ≤ C (cid:18) ˆ B j ˆ B j max { w j ( x ) , w j ( y ) } p | ψ j ( x ) − ψ j ( y ) | p dµ + ˆ B j w pj |∇ ψ | p dx + ess sup x ∈ supp ψ j ˆ R n \ B j w j ( y ) p − | x − y | n + ps dy · ˆ B j w j ψ pj dx (cid:19) , (9.2)where C = C ( n, p, s, Λ). By using the fact that ¯ k j ≤ k j along with the properties of ψ j in(9.2), we obtain ˆ ¯ B j |∇ w j | p dx ≤ C ( n, p, s, Λ)( I + I + I ) , (9.3)where I = ˆ B j ˆ B j max { w j ( x ) , w j ( y ) } p | ψ j ( x ) − ψ j ( y ) | p dµ,I = ˆ B j w pj |∇ ψ j | p dx and I = ess sup x ∈ supp ψ j ˆ R n \ B j w j ( y ) p − | x − y | n + ps dy · ˆ B j w j ψ pj dx. Let A j = { u < k j } ∩ B j . We estimate the terms I j , for j = 1 , ,
3, separately.
Estimate of I : Using the fact that r < r j < r , w j ≤ M , r ∈ (0 ,
1] and the properties of ψ j ,we obtain I = ˆ B j ˆ B j max { w j ( x ) , w j ( y ) } p | ψ j ( x ) − ψ j ( y ) | p dµ ≤ C ( n, p, s, Λ) 2 jp r p M p | A j | . (9.4) ixed local and nonlocal quasilinear equations Estimate of I : Using the properties of ψ j and the fact that w j ≤ M , we have I = ˆ B j w pj |∇ ψ j | p dx ≤ C ( n, p, s, Λ) 2 jp r p M p | A j | . (9.5) Estimate of I : For every x ∈ supp ψ j and every y ∈ R n \ B j , we observe that1 | x − y | = 1 | y | | x − ( x − y ) || x − y | ≤ | y | (1 + 2 j +3 ) ≤ j +4 | y | . (9.6)Then using r j > r , w j ≤ M , 0 ≤ ψ j ≤ µ − ≤ ess inf R n u, we obtain I = ess sup x ∈ supp ψ j ˆ R n \ B j w j ( y ) p − | x − y | n + ps dy · ˆ B j w j ψ pj dx ≤ C ( n, p, s ) 2 j ( n + ps ) r p M p | A j | , (9.7)for every r ∈ (0 , ˆ B j |∇ w j | p dx ≤ C ( n, p, s, Λ) 2 j ( n + ps + p ) r p M p | A j | . (9.8)Noting that B j +1 ⊂ ¯ B j ⊂ B j , ¯ w j ≤ w j and using the Sobolev inequality in (2.2), we obtain(1 − a ) M j +2 | A j +1 | = ˆ A j +1 (¯ k j − k j +1 ) dx ≤ ˆ B j +1 ¯ w j dx ≤ ˆ B j w j dx ≤ | A j | − pκ (cid:16) ˆ B j w pκj ψ pκj dx (cid:17) pκ ≤ Cr npκ − np | A j | − pκ (cid:16) ˆ B j |∇ ( w j ψ j ) | p dx (cid:17) p (9.9)for a constant C = C ( n, p, s ) and κ as given in (2.1). By using (9.8) together with the factthat w j ≤ M and the properties of ψ j in (9.9), we get1 − a j +2 | A j +1 | ≤ Cr np ( κ − j ( n + ps + p ) p | A j | − pκ + p . (9.10)Since r < ¯ r j < r , we obtain from (9.10) that | A j +1 | ≤ Cr np ( κ − − a j (2+ n + psp ) | A j | − pκ + p . (9.11)By dividing both sides of (9.11) with | B j +1 | and noting that | B j | < n | B j +1 | together with r j < r , we obtain Y j +1 ≤ C ( n, p, s, Λ)1 − a j (2+ n + psp ) Y p (1 − κ ) j , (9.12)where we denote Y j = | A j || B j | , j = 0 , , , . . . . By choosing c = C ( n, p, s, Λ)1 − a , b = 2 (2+ n + psp ) , β = 1 p (cid:16) − κ (cid:17) , τ = c − β b − β ∈ (0 , Y j → j → ∞ , if Y ≤ τ . This implies that u ≥ µ − + aM almosteverywhere in B r (0). ixed local and nonlocal quasilinear equations References [1] Siva Athreya and Koushik Ramachandran. Harnack inequality for non-local Schr¨odingeroperators.
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