SUPERPOSITION IN THE P-LAPLACE EQUATION. KARL K. BRUSTAD Abstract. That a superposition of fundamental solutions to the p-Laplace Equation is p-superharmonic – even in the non-linear 6 cases p > 2 – has been known since M. Crandall and J. Zhang 1 published their paper Another Way to Say Harmonic in2003. We 0 give a simple proof and extend the result by means of an explicit 2 formula for the p-Laplacian of the superposition. n a J 8 1 1. Introduction ] Our object is a superposition of fundamental solutions for the p- P Laplace Equation A . (1.1) ∆ u := div |∇u|p−2∇u = 0. h p at Although the equation is non-lin(cid:0)ear, the fun(cid:1)ction m ρ(y) [ V(x) = dy, ρ ≥ 0, 2 ≤ p < n 1 ZRn |x−y|np−−1p v is a supersolution in Rn, i.e. ∆ V ≤ 0 in the sense of distributions. It p 2 is a so-called p-superharmonic function – see Definition 2 on page 6 – 9 4 according to which it has to obey the comparison principle. The case 4 p = 2 reduces to the Laplace Equation ∆u = 0 with the Newtonian 0 potential . 1 ρ(y) 0 V(x) = dy, 6 |x−y|n−2 Rn 1 Z which is a superharmonic function. : v M. Crandall and J. Zhang discovered in [CZ03] that the sum i X N a r i , a > 0 a n−p i i=1 |x−yi|p−1 X of fundamental solutions is a p-superharmonic function. Their proof was written in terms of viscosity supersolutions. A different proof was given in [LM08]. The purpose of our note is a simple proof of the following theorem: Date: January 18, 2016. 1 2 KARL K.BRUSTAD Theorem 1. Let 2 ≤ p < n. For an arbitrary concave function K, ∞ a (1.2) W(x) := i +K(x), y ∈ Rn, a ≥ 0, n−p i i i=1 |x−yi|p−1 X is p-superharmonic in Rn, provided the series converges at some point. Through Riemann sums one can also include potentials like ρ(y) dy +K(x), ρ ≥ 0. n−p ZRn |x−yi|p−1 Similar results are given for the cases p = n and p > n and, so far as we know, the extra concave term K(x) is a new feature. The key aspect of the proof is the explicit formula (3.2) for the p-Laplacian of the superposition. Although the formula is easily obtained, it seems to have escaped attention up until now. Finally, we mention that in [GT10] the superposition of fundamental solutions has been extended to the p-Laplace Equation in the Heisen- berg group. (Here one of the variables is discriminated.) In passing, we show in Section 6 that similar results are not valid for the evolutionary equations ∂ ∂ u = ∆ u and (|u|p−2u) = ∆ u p p ∂t ∂t where u = u(x,t). We are able to bypass a lenghty calculation in our counter examples. 2. The fundamental solution Consider a radial function, say f(x) = v(|x|) where we assume that v ∈ C2(0,∞). By differentiation v′ (2.1) ∇f = xT, |∇f| = |v′|, |x| xxT v′ xxT v′ Hf = v′′ + I − , ∆f = v′′ +(n−1) , |x|2 |x| |x|2 |x| (cid:18) (cid:19) when x 6= 0. The Rayleigh quotient formed by the Hessian matrix Hf = ∂2f ∂xi∂xj above will play a central role. Notice that for any non-zero z ∈hRn, wie have that zT xxT z = cos2θ |z| |x|2 |z| 3 where θ is the angle between the two vectors x and z. This yields the expedient formula zT(Hf)z v′ (2.2) = v′′cos2θ+ sin2θ, x,z 6= 0. |z|2 |x| Since the gradient of a radial function is parallel to x, the Rayleigh quotient in the identity ∇f(Hf)∇fT (2.3) div |∇f|p−2∇f = |∇f|p−2 (p−2) +∆f |∇f|2 (cid:18) (cid:19) (cid:0) (cid:1) reduces to v′′. The vanishing of the whole expression is then equivalent to v′ (2.4) (p−1)v′′ +(n−1) = 0 |x| which, integrated once, implies that a radially decreasing solution w is on the form (2.5) w(x) = v(|x|) where v′(|x|) = −c|x|1p−−n1. The constant c = c > 0 can now be chosen so that n,p ∆ w +δ = 0 p in the sense of distributions. Thus (2.6) w(x) = −cn,ppp−−n1|x|pp−−n1, when p 6= n, (−cn,nln|x|, when p = n is the fundamental solution to the p-Laplace Equation (1.1). 3. Superposition of fundamental solutions We now form a superposition of translates of the fundamental solution and compute its p-Laplacian. To avoid convergence issues all sums are, for the moment, assumed finite. Lemma 1. Let w be the fundamental solution to the p-Laplace equation. Define the function V as N (3.1) V(x) := a w(x−y ), a > 0, y ∈ Rn. i i i i i=1 X Then, in any dimension and for any p 6= 1 1, ∆ V is of the same sign p wherever it is defined in Rn. Furthermore, the dependence of the sign on p and n is as indicated in figure 1. 1When p = 1 there are no non-constant radial solutions of (1.1). Instead we get the zero mean curvature equation in which a solution’s level sets are minimal surfaces. 4 KARL K.BRUSTAD n 4 3 2 1 −2 −1 0 1 2 3 4 p Figure 1. ∆ V ≤ 0, ∆ V = 0, ∆ V ≥ 0 p p p Proof. We simplify the notation by letting w and v denote that the i i functions w and v are to be evaluated at x−y and|x−y |, respectively. i i First, the linearity of the Hessian and the Laplacian enable us to write ∇V(HV)∇TV ∆ V = |∇V|p−2 (p−2) +∆V p |∇V|2 (cid:18) (cid:19) N ∇V(Hw )∇TV = |∇V|p−2 a (p−2) i +∆w . i |∇V|2 i i=1 (cid:18) (cid:19) X Secondly, by (2.1) and (2.2) this is N v′ = |∇V|p−2 a (p−2) v′′cos2θ + i sin2θ i i i |x−y | i Xi=1 (cid:18) (cid:16) i (cid:17) v′ + v′′ +(n−1) i i |x−y | i (cid:19) N v′ = |∇V|p−2 a (p−2) i −v′′ sin2θ i |x−y | i i Xi=1 (cid:18) (cid:16) i (cid:17) v′ + (p−1)v′′ +(n−1) i i |x−y | i (cid:19) where θ is the angle between x−y and ∇V(x). And finally, as w is i i a fundamental solution, the last two terms disappear by (2.4). We get N v′ ∆ V = (p−2)|∇V|p−2 a i −v′′ sin2θ . p i |x−y | i i i=1 (cid:18) i (cid:19) X 5 It only remains to use the formula (2.5) for v′ to compute that i |x−vi′y | −vi′′ = −cn,pp+p−n−1 2|x−yi|2−p−n−1p i and the sign of ∆ V can easily be read off the final identity p N sin2θ (3.2) ∆ V(x) = −c (p−2)(p+n−2)|∇V|p−2 a i . p n,p p−1 i p+n−2 i=1 |x−yi| p−1 X (cid:3) Remark 1. Thethreegreenlinesinfigure1deservesomeattention. The line p = 2 is obvious since the equation becomes linear. So is the line n = 1 asthe“angle”between two numbers is0 orπ. The littlesurprise, perhaps, is the case p + n = 2. Then the terms in V will be on the form a |x−y |2 and it all reduces to the rather unexciting explanation i i that a linear combination of quadratics is again a quadratic. 4. Adding more terms We will now examine what will happen to the sign of the p-Laplace operator when an extra term, K(x), is added to the linear combination (3.1). We will from now on only consider p > 2. Restricted to this case, the factor C := c (p−2)(p+n−2) in (3.2) stays positive. n,p n,p p−1 Let V be as in Lemma 1 and let K ∈ C2. For efficient notation, write ξ = ξ(x) := ∇V(x)+∇K(x). Then ξH(V +K)ξT ∆ (V +K) = |ξ|p−2 (p−2) +∆(V +K) p |ξ|2 (cid:18) (cid:19) ξ(HV)ξT = |ξ|p−2 (p−2) +∆V |ξ|2 (cid:18) (cid:19) ξ(HK)ξT +|ξ|p−2 (p−2) +∆K . |ξ|2 (cid:18) (cid:19) Now, the second to last term equals −Cn,p|ξ|p−2 ai|x−yi|2−p−n−1p sin2αi ≤ 0 i X where α is the angle between x − y and ∇V(x) + ∇K(x). Thus it i i suffices to ensure that the last term also is non-positive in order for the p-Laplace to hold its sign. Lemma 2 presents a sufficient condition. Lemma 2. Let p > 2 and define V as in (3.1). Then (4.1) ∆ (V(x)+K(x)) ≤ 0 p for all concave functions K ∈ C2(Rn) wherever the left-hand side is defined. 6 KARL K.BRUSTAD Proof. zT(HK)z ≤ 0 for all z ∈ Rn since the Hessian matrix of a concave function K is negative semi-definite. Also K is superharmonic since the eigenvalues of HK are all non-positive, i.e. ∆K ≤ 0. There- fore, ξ(HK)ξT ∆ (V(x)+K(x)) ≤ |ξ|p−2 (p−2) +∆K ≤ 0. p |ξ|2 (cid:18) (cid:19) (cid:3) Remark 2. Though K ∈ C2 being concave is sufficient, it is not neces- sary. A counter example is provided by the quadratic form 1 K(x) = xTAx, where A = diag(1−m,1,...,1), m = p+n−2. 2 ThenK isnotconcave, butacalculationwillconfirmthat(p−2)ξ(HK)ξT+ |ξ|2 ∆K ≤ 0 and hence ∆ (V + K) ≤ 0. In fact, a stronger result than p Lemma 2 is possible: Let f be C2 at x for i = 1,...,N and let i λi ≤ λi ≤ ··· ≤ λi 1 2 n be the eigenvalues of the Hessian matrix Hf (x). If i λi +···+λi +(p−1)λi ≤ 0 ∀i, 1 n−1 n then ∆ ( f ) ≤ 0 at x. p i i P 5. p-superharmonicity We now prove that ∞ W(x) := a w(x−y )+K(x), a ≥ 0, y ∈ Rn, K concave i i i i i=1 X is a p-superharmonic function in Rn. The three cases 2 < p < n, p = n and p > n are different and an additional assumption, (5.3), seems to be needed when p ≥ n. In the first case, only convergence at one point is assumed. We start with the relevant definitions and a useful Dini-type lemma. Definition 1. Let Ω be a domain in Rn. A continuous function h ∈ W1,p(Ω) is p-harmonic if loc (5.1) |∇h|p−2∇h∇φT dx = 0 Z for each φ ∈ C∞(Ω). 0 Definition 2. A function u: Ω → (−∞,∞] is p-superharmonic in Ω if i) u 6≡ ∞. ii) u is lower semi-continuous in Ω. iii) If D ⊂⊂ Ω and h ∈ C(D) is p-harmonic in D with h ≤ u , ∂D ∂D then h ≤ u in D. (cid:12) (cid:12) (cid:12) (cid:12) 7 Furthermore, if u ∈ C2(Ω), it is a standard result that u is p- harmonic if and only if ∆ u = 0 and u is p-superharmonic if and p only if ∆ u ≤ 0. p Also, a function u in C(Rn)∩W1,p(Rn) is p-superharmonic if loc (5.2) |∇u|p−2∇u∇φT dx ≥ 0 Rn Z for all 0 ≤ φ ∈ C∞(Rn). See [Lin86]. 0 Lemma 3. Let(f ) be anincreasingsequenceof lowersemi-continuous N (l.s.c.) functions defined on a compact set C converging point-wise to a function f ≥ 0. Then, given any ǫ > 0 there is an N ∈ N such that ǫ f (x) > −ǫ N for all x ∈ C and all N ≥ N . ǫ The standard proof is omitted. In the following, K is any concave function in Rn. We let K , δ > 0 δ denote the smooth convolution φ ∗K with some mollifier φ . One can δ δ show that K is concave and δ K → K δ locally uniformly on Rn as δ → 0+. 5.1. The case 2 0. If y ∈ Rn and a > 0, the function i i N a Wδ(x) := i +K (x) N n−p δ i=1 |x−yi|p−1 X is p-superharmonic except possibly at the poles y (Lemma 2). Defin- i ing Wδ(y ) := ∞, we claim that Wδ is p-superharmonic in the N i N whole Rn. We have to verify Def. 2. Clearly, i) and ii) are valid. For the comparison principle in iii) we select D ⊂⊂ Rn (i.e. D is bounded) and let h ∈ C(D) be p-harmonic in D with h ≤ Wδ . If any, ∂D N ∂D isolate the points y in D with ǫ-balls B := B(y ,ǫ) where ǫ > 0 is so i i (cid:12)i (cid:12) small so that Wδ ≥ max h. This is possible(cid:12)because h(cid:12)is bounded N Bi D and because lim Wδ(x) = ∞. Then Wδ is C2 on D \ ∪B so, by x→(cid:12)yi N N i Lemma 2, ∆ W (cid:12)≤ 0 on this set. Also, h ≤ Wδ by p N ∂(D\∪Bi) N ∂(D\∪Bi) the construction of the ǫ-balls, so h ≤ Wδ on this set since Wδ is (cid:12)N (cid:12) N p-superharmonic there. Naturally, h ≤ Wδ(cid:12) on ∪B , so the(cid:12) inequality N i will hold in the whole domain D. This proves the claim. Now N → ∞. Assume that the limit function ∞ a Wδ(x) := i +K (x) n−p δ i=1 |x−yi|p−1 X 8 KARL K.BRUSTAD is finite at least at one point in Rn. We claim that Wδ is p- superharmonic. By assumption Wδ 6≡ ∞ and it is a standard result that the limit of an increasing sequence of l.s.c functions is l.s.c. Part iii). Suppose that D ⊂⊂ Rn and h ∈ C(D) is p-harmonic in D with h ≤ Wδ . Then (Wδ −h) is an increasing sequence of l.s.c. ∂D ∂D N functionsonthecompactset ∂D withpoint-wiselimit(Wδ−h) ≥ 0. (cid:12) (cid:12) ∂D If ǫ > 0(cid:12), then (W(cid:12)δ −h) > −ǫ for a sufficiently big N by Lemma 3. N ∂D (cid:12) That is (cid:12) (cid:12) (cid:12)(h−ǫ) < Wδ ∂D N ∂D so(h−ǫ) ≤ Wδ sinceh−ǫis(cid:12)p-harmoni(cid:12)candWδ isp-superharmonic. D N D (cid:12) (cid:12) N Finally, since Wδ ≤ Wδ we get (cid:12) N(cid:12) (cid:12) (cid:12) (h−ǫ) ≤ Wδ D D and as ǫ was arbitrary, the require(cid:12)d inequa(cid:12)lity h ≤ Wδ in D is obtained (cid:12) (cid:12) and the claim is proved. Let δ → 0 and set ∞ a i W(x) := +K(x). n−p i=1 |x−yi|p−1 X We claim that W is p-superharmonic. Part i) and ii) are immediate. For part iii), assume D ⊂⊂ Rn and h ∈ C(D) is p-harmonic in D with h ≤ W . Let ǫ > 0. Then ∂D ∂D there is a δ > 0 such that (cid:12) (cid:12) (cid:12) (cid:12) |K(x)−K (x)| < ǫ δ at every x ∈ D. We have Wδ = W +K −K > W −ǫ ≥ h−ǫ δ on∂D. Andagain,sinceh−ǫisp-harmonicandWδ isp-superharmonic, we get Wδ ≥ h−ǫ in D. Thus W ≥ Wδ −ǫ ≥ h −2ǫ. D D D This proves the claim,(cid:12)settles t(cid:12)he case 2 <(cid:12) p < n and completes the (cid:12) (cid:12) (cid:12) proof of Theorem 1. We now turn to the situation p ≥ n and introduce the assumption ∞ (5.3) A := a < ∞. i i=1 X 9 5.2. The case p=n. Let δ > 0. The partial sums N Wδ(x) := − a ln|x−y |+K (x) N i i δ i=1 X are p-superharmonic in Rn by the same argument as in the case 2 < p < n. Let N → ∞. We claim that ∞ Wδ(x) := − a ln|x−y |+K (x) i i δ i=1 X is p-superharmonic in Rn provided the sum converges absolutely2 at least at one point. Assume for the moment that, given a radius R > 0, it is possible to find numbers C so that i ln|x−y | ≤ C for all x ∈ B := B(0,R),and i i R ∞ (5.4) the series a C =: S converges. i i R i=1 X Define the sequence (f ) in B by N R N f (x) := −a ln|x−y |+a C +K (x), f(x) := lim f (x). N i i i i δ N N→∞ i=1 X(cid:0) (cid:1) Then (f ) is an increasing sequence of l.s.c functions implying that f N is l.s.c. in B and that R Wδ = f −S R is as well. Since R can be arbitrarily big, we conclude that Wδ does not take the value −∞ and is l.s.c. in Rn. For part iii) we show that f obeys the comparison principle. Assume D ⊂⊂ B and h ∈ C(D) is p-harmonic in D with h ≤ f . Then R ∂D ∂D (f − h) is an increasing sequence of l.s.c. functions on the compact N (cid:12) (cid:12) set ∂D with point-wise limit (cid:12) (cid:12) (f −h) ≥ 0. ∂D If ǫ > 0, then (f −h) > −ǫ fo(cid:12)r a sufficiently big N by Lemma 3. N ∂D (cid:12) That is (cid:12) (cid:12) (h−ǫ) < f ∂D N ∂D so(h−ǫ) D ≤ fN D sinceh−ǫisp(cid:12)(cid:12)-harmoni(cid:12)(cid:12)candfN isp-superharmonic. Finally, since f ≤ f we get N (cid:12) (cid:12) (cid:12) (cid:12) (h−ǫ) ≤ f D D (cid:12) (cid:12) 2Conditional convergence is not suffi(cid:12)cient. A(cid:12) counter example is ai = 1/i2, i δ |yi|= exp((−1) i), yielding W (x)=−∞ for all yi 6=x6=0. 10 KARL K.BRUSTAD and as ǫ was arbitrary, the required inequality h ≤ f in D is obtained. Hence Wδ(x) = f(x) − S is a p-superharmonic function in any ball R B . R The claim is now proved if we can establish the existence of the numbers C satisfying (5.4). By a change of variables we may assume i that the convergence is at the origin. That is ∞ L := a |ln|y || < ∞. i i i=1 X We have ln|x−y | ≤ ln(|x|+|y |) i i ≤ ln(2max{|x|,|y |}) i = max{ln|x|,ln|y |}+ln2, i so C := max{lnR,ln|y |}+ln2 i i will do since (for R > 1/2) the sequence of partial sums N a C is i=1 i i increasing and bounded by Aln2R+L. P The final limit δ → 0 causes no extra problems. ∞ W(x) := − a ln|x−y |+K(x) i i i=1 X is p-superharmonic in Rn. This settles the case p = n. 5.3. The case p>n. Let δ > 0. Consider again the partial sums N p−n WNδ(x) := − ai|x−yi|p−1 +Kδ(x). i=1 X As before Wδ is p-superharmonic in Rn, but now a different N approach is required for the proof. For ease of notation, write N p−n u(x) := − a |x−y |α +K(x), 0 < α := < 1, i i p−1 i=1 X where K ∈ C∞(Rn) is concave. We will show that u satisfies the integral inequality (5.2). Clearly, u is continuous and |u|pdx < ∞ on any bounded domain Ω Ω. Also, R xT p 1 |∇(|x|α)|p = α ∝ |x|2−α |x|(1−α)p (cid:12) (cid:12) where one can show that (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (1−α)p < n. Thus |∇u|pdx < ∞ locally so u ∈ C(Rn)∩W1,p(Rn). loc R

Superposition in the $p$-Laplace Equation

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