Removable singularities for some degenerate non-linear elliptic equations

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1 Mathematica Aeterna, Vol. 5, 2015, no. 1, Removable singularities for some degenerate non-linear ellitic equations Tahir S. Gadjiev Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9, B.Vahabzade Str., AZ 1141, Baku, Azerbaijan. Nigar R. Sadykhova Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9, B.Vahabzade Str., AZ 1141, Baku, Azerbaijan. Abstract This aer is devoted to studying the removable singularities of solutions for the Dirichlet roblem for degenerate non-linear ellitic equations on the boundary of domain. Method a riori energetic estimates of solutions to ellitic boundary value roblems is used. The alied method differs from the way for obtaining aroriate results in linear situation. Mathematics Subject Classification: 32D20, 35J70 Keywords: removable singularities, degenerate, ellitic equation 1 Introduction The corresonding results for linear equations were obtained in the aers of L.Carleson [1], V.A.Kondratyev, O.A.Oleynik [2], O.A.Oleynik, G.A.Iosifyan [3], V.A.Kondratyev, E.M.Landis [4], D.Gilbarg, N.Trudinger [5], T.Gadjiev, V.Mamedova [6], J.Diederich [7], R.Harvey, J.Polking [8], for non-linear equations in the aers of T.Kilelainen, X.Zhong [9] and others. Let Ω R n, n 2 be a bounded domain. Consider the following equation ( 1) α D α A α (x, u, Du,..., D m u) = ( 1) α D α F α (x), (1)

2 22 Tahir S. Gadjiev and Nigar R. Sadykhova where D α = α x α 1 1 x α x αn n, α = α 1 + α α n, m 1. Assume that the coefficients A α (x, ξ) of the equation (1) are measurable with resect to x Ω, are continuous with resect to ξ R M (M is the number of different multi-indexes of lengths no more than m) and satisfy the conditions α =m A α (x, ξ)ξ m α ω(x) ξ m m 1 c 1 ω(x) ξ i f 1 (x), i=1 m A α (x, ξ) c 2 ω(x) ξ i 1 + f2 (x), (2) i=0 where ξ = (ξ 0,..., ξ m ), ξ i = (ξ i α), α = i, > 1, f 1 (x) L 1,loc (Ω), f 2 (x) L 1,loc (Ω), F α L 1,loc (Ω). Suose that ω(x), x Ω is a measurable non-negative function satisfying the conditions: ω L 1,loc (Ω), and for any > 0 and some σ > 1 ω 1/(σ 1) dx <, ess su ω(x) c 3 ρ n(σ 1) x Ω ρ Ω ρ Ω ρ 1 σ ω 1 (σ 1) dx. (3) Here Ω ρ = Ω B ρ, B ρ = {x : x < ρ}, c i are ositive constants deendent only on the roblem data. In articular, it follows from conditions (3) that ω A σ ([10]), i.e. σ 1 ωdx ω 1 σ 1 dx c 4 ρ nσ. (4) Ω ρ Ω ρ The following estimate also follows from (3) ess su ω(x) c 5 ρ n x Ω ρ Ω ρ ωdx. (5) Furthermore, assume that ( ) ω(ω s ) s nµ ω(ω h ) c 6, (6) h µ < 1 + /n, for any s h > 0, where ω(ω s ) = Ω s ω(x)dx.

3 Removable singularities 23 2 Some definitions and auxiliary results We ll describe geometry of Ω by means of the non-linear basic frequency λ (r) of the cross section S r 1 λ (r) = inf S v ds v ds, (7) S r where the lower bound is taken on all continuously differentiable in some vicinity of S r functions vanishing on Ω; S v(x) is a rojection of the vector v(x) on a tangential lane to S r at the oint x. For = 2 the number λ 2 2(r) is the first eigenvalue of Beltrami-Lalace oerator on S r, for 2 λ (r) was studied in various aers. Some examles of calculation, or its lower estimates for a number of secific sets are for examle in [11]. By W,ω(Ω) m we denote a closure of the functions from C m ( Ω) with resect to the norm 1 u W m,ω (Ω) = ω(x) D α u dx Ω W m,ω is a closure of the functions from C0 (Ω) to W,ω(Ω). m We say that the function u(x) W m,ω(ω) is a generalized solution of the Dirichlet roblem for equation (1) if the following integral identity is fulfilled for an arbitrary function η(x) C0 (Ω) A α (x, u,..., D m u)d α ηdx = F α (x)d α ηdx. (8) Ω S r Ω We ll divide the considered domains into two classes. The first class is narrow domains whose comlement in the vicinity of the oint 0 is sufficiently massive, for examle it contains some cone with a vertex at this oint. In the terms of frequency of set this class of domains satisfies the condition A)rλ (r) > d 1 > 0, r (0, r 0 ), r 0 > 0. The second class contains wide domains, i.e. such that have inwards cus at the oint 0. In the terms of frequency of the set this class of domains is described as following B)rλ (r) < d 2 <, r (0, r 0 ). Determine the function ψ(r) on (0, r 0 ) by the inequality inf λ ( x )(r rψ(r))ω(x) µ > 0, (9) rψ(r)< x <r where µ is such that 0 < 1 c 0 < ψ(r) < 1. For monotonically decreasing functions λ (r) (we meet them in alications) inequality (9) accets the following form rλ (r)(1 ψ(r))ω(x) µ, for ϕ(r) 1 ψ(r) µω 1 (x)(rλ (r)) 1. (10)

4 24 Tahir S. Gadjiev and Nigar R. Sadykhova Consider the distance function from the oint x to Ω g (x) = ρ (x, Ω). It is known that δ > 0 such that Γ δ = {x : 0 < ρ (x, Ω) < δ}, g (x) C m, g (x) = 1. Furthermore, it follows from [5] that j g (x) h 0 (g (x)) 1 j, x Γ δ, j = 1, m. (11) Denote Ω r = Ω {x : g (x) < r} For an arbitrary Γ Ω by W m,ω(ω, Γ) we denote a closure in the norm W m,ω(ω) of the set of functions from C (Ω) vanishing near Ω\Γ. We ll say that u(x) W m,ω,loc(ω, Γ) if u(x) W m,ω(ω, Ω \ Ω) for any subdomain Ω Ω such that Γ Ω =. Let u(x) W m,ω(ω, Γ) be a generalized solution of equation (1), i.e. u (x) satisfy integral identity (8) for any function η (x) C 0 (Ω ), Ω Ω, Γ Ω =. Formulate some auxiliary lemmas. Lemma 2.1 Let I (r) be a non-negative non-increasing on the interval (0, r 0 ), r 0 > 0 function satisfying the condition I (r) < θi (rε (r)) + G (rε (r)), 0 < θ < 1, (12) where ε (r) is a measurable function, 0 < c 0 < ε (r) < 1 is such that K (r) (ϕ (r)) 1 rε (r) < τ < rinf ϕ (τ) ν > 0, ϕ (r) 1 ε (r), (13) and G (r) is measurable and locally bounded. Then the following alternative is valid. Either I (r i ) < c 7 G (r i ) for some sequence r i 0, or I (r) sufficiently raidly grows as r 0, exactly r 0 I (r) c 7 ex c 8 ν ln (θ + δ) 1 dτ I (r 0 ), 0 < δ < 1 θ, (14) τ (1 ε (τ)) r where c 7, c 8 > 0 are constants. In articular, the last estimate also holds in the case of boundedness of G (r) for any unbounded function I (r) satisfying condition (12). Lemma 2.2 Let I (r) be a non-negative non-increasing function on the interval (0, r 0 ) satisfying the condition I (r) (1 ϕ (r)) I (rε) + ϕ (r) G (rε), 0 < ε < 1, r (0, r 0 ), (15) where ϕ (r) is a measurable function and 0 < ϕ (r) < c 0 < 1, r (0, r 0 ). G (r) is measurable and locally bounded. Then the following alternative is valid for I (r).

5 Removable singularities Either for some sequence r i 0 the estimate I (r i ) < c 9 G (r i ), c 9 < is fulfilled; 2. or I (r) raidly grows as r 0, exactly, I (r) c 10 ex 1 δ ln ε 1 r 0 r ϕ (τ) dτ τ I (r 0 ), δ > 0, r (0, r 0 ), r 0 = r 0 (δ), (16) where ϕ (r) is an arbitrary continuous, non-decreasing function satisfying the inequality ϕ (r) ϕ (r), r (0, r 0 ). 3 The behaviour of integral energy Now we study behaviour of I (r) for small r. Theorem 3.1 Let u(x) W m,ω,loc(ω, Γ) be a generalized solution of the Dirichlet roblem for equation (1). Suose that the coefficients of the equation satisfy condition (2), the domain Ω satisfies the condition A). ψ (t) be a measurable function satisfying conditions (9), and let for the function K (r) estimate B) be fulfilled with resect to ϕ (r) = 1 ψ (r). Then the following alternative is valid for I (r): 1. Either I (r i ) < c 11 (1 + G (r i )) for some sequence r i 0, where c 11 < is a constant; 2. or I (r) grows raidly as r 0, exactly r 0 I (r) > c 12 (γ) ex c 0 ν ln (k 0 + γ) 1 dτ, r < r 0 = r 0 (γ), (17) τϕ (τ) where k 0 is some constant. Proof. Substitute the test function η (x) = u (x) ( 1 ξ ψ(r) (r 1 g (x)) ) into integral identity (17). For simlicity, we ll assume that the solution u (x, t) is sufficiently smooth. Therefore we admit some formality in reasonings. Since, these reasonings may be recise by assing to a regularized roblem by the known method (see [12]) and later tending the regularization arameter to zero we ll obtain the result for generalized solution. Continuing the roof of the theorem, we use condition (2) and get Ω\Ω r r ω (x) D m u dx

6 26 Tahir S. Gadjiev and Nigar R. Sadykhova Ω\Ω rψ(r) + Ω r0 \Ω rψ(r) + c 3 k 1 ω (x) + 1 Ω\Ω r0 β α k 2 ω (x) D α u 1 β < α f 2 (x) D α β u D β ξ dx+ 1 D α u + k 2 ω (x) 1 α <m 1 D α u + D α u + k 1 ω (x) D α β u D β ξ + D α u 1 f 2 (x) D α u + f 1 (x) ξdx+ D α u + k 1 ω (x) α <m D α u 1 f 2 (x) D α u + f 1 (x) ξdx. (18) Notice that we are in the class of domains for which λ (r) as r 0, consequently for any δ > 0 there exists r 0 = r 0 (δ) such that for any r < r 0 λ (r) > δ 1. Using the Young inequality with ε, from (18) we get ( ) I (r) [I (rψ (r)) I (r)] Ω r\ω rψ(r) Ω\Ω rψ(r) ( k 3 (1 δ ) 1 + ε + ( 1) ε 1 1 k4 f 2 (x) m α 1 λ 1 (g (x)) dx + [I (rψ (r)) I (r 0 )] k 5 + ε α <m ) + I (r 0 ) ( k 6 + ε f 2 (x) m α 1 λ 1 ) + ( 1) ε 1 1 (g (x)) dx + f 1 (x) dx. (19) Denote G (r) [ f 2 (x) m α 1 λ 1 (g (x)) dx+ f 1 (x) ] dx. Ω\Ω r Then from (19) we get I (r) α (δ, ε) I (rψ (r)) + A 1 I (r 0 ) + B 1 G (rψ (r)), (20) where α, A 1, B 1 are some constants that are exactly calculated and α 0 = α (0, 0) < 1. Thus, from (20) and Lemma 2.1 we get the roof of the theorem.

7 Removable singularities 27 References [1] Carleson L. Selected roblems on excetional sets. D.Van. Nostrand comany, Toronto-London-Melbourne, 1967, 126. [2] Kondratyev V.A., Oleynik O.A. Boundary value roblems of mathematical hysics and related roblems 14 (za. nauch. sem. LOMI, vol. 115), Nauka, 1982, (in Russian). [3] Oleynik O.A., Iosifyan G.A. On excetional singularities on a boundary and uniqueness of solutions of boundary value roblems for second order ellitic and arabolic equations. Funk. Anal., 1977, v.ii, issue 3, (in Russian). [4] Kondratyev V.A., Landis E.M. Qualitative theory of artial differential equations of second order. Itogi nauki i tekhniki. Ser. Modern roblems of mathematics, v.3, 1988, (in Russian) [5] Gilbarg D., Trudinger N. Ellitic artial differential equations of second order. Berlin-New-York, Sringer-Verlag, 1977, 401. [6] Gadjiev T., Mamedova V. On removable sets of solutions of the second order ellitic and arabolic equations in nondivergent form. Ukr. Math. Journ., 2009, v.61, No 11, [7] Diederich J. Removable singularities of solutions of ellitic artial differential equations. Trans.Amer. Math. Soc., 165, 1972, [8] Harvey R., Polking J. Removable singularities of solutions of linear artial differential equations. Acta Math. 125, 1970, [9] Kilelainen T., Zhong X. Removable sets for continuous solutions of quasilinear ellitic equations. Proc. Amer. Math. Soc. 2002, 130, No 6, [10] Chanillo S., Wheeden R. Weighted Poincare and Sobolev inequalities and estimates for weighted Peano maximal functions. Amer. J.Math., 1985, 107, [11] Milyukov V.M. On asymtotic roerties of subsolutions of ellitic tye quasilinear equations. 1980, v 3(145), No [12] Alikakos N.D., Rostamian R. Gradient estimates for degenerate diffusion equations. Math. Ann. 1982, v.259, No 1, Received: November, 2014