“Neumann𝑝-Laplacian problems with a reaction term on metric spaces”

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3rdBYMAT Conference(2020)

N e u m a n n

𝑝

- L a p l a c i a n p r o b l e m s w i t h a r e a c t i o n t e r m o n m e t r i c s p a c e s

� A n t o n e l l a N a s t a s i

A b s t r a c t : We use a variational approach to study existence and regularity of solu-

tions for a Neumann𝑝-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.

R e s u m e n : Utilizamos un enfoque variacional para estudiar la existencia y regula-

ridad de soluciones para un problema de Neumann𝑝-Laplaciano con un término de reacción en espacios métricos dotados de una medida de duplicación y que per- miten una desigualdad de Poincaré. Se aplican teoremas de traza para funciones con variación acotada en la definición del funcional variacional y se demuestra que los minimizadores satisfacen condiciones de tipo De Giorgi.

K e y w o r d s : 𝑝-Laplacian operator, measure metric spaces, minimal𝑝-weak upper

gradient, minimizer.

M S C 2 0 1 0 : 31E05, 30L99, 46E35.

R e f e r e n c e : NASTASI, Antonella. “Neumann𝑝-Laplacian problems with a reaction term on metric spaces”. In:

TEMat monográficos, 2 (2021):Proceedings of the 3rd BYMAT Conference, pp. 87-90. ISSN: 2660-6003. URL:

https://temat.es/monograficos/article/view/vol2-p87.

cb This work is distributed under a Creative Commons Attribution 4.0 International licence https://creativecommons.org/licenses/by/4.0/

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Neumann𝑝-Laplacian problems with a reaction term on metric spaces

1 . I n t r o d u c t i o n

We extend existence and regularity results for a Neumann boundary value problem valid on the Euclidean setting and, more generally, in Riemannian manifolds (see Nastasi [8]) to the general setting of metric spaces. Applying variational methods such as those based on De Giorgi classes [2], we study a Neumann boundary value problem as in Lahti, Malý, and Shanmugalingam [5] and Malý and Shanmugalingam [6], but the new feature is that we include a reaction term (see Nastasi [9]). Under appropriate conditions on the reaction term, we prove existence and boundedness properties of solutions with a reaction term in a metric space equipped with a doubling measure and supporting a Poincaré inequality and thus extending the corresponding results in Kinnunen and Shanmugalingam [3] and Malý and Shanmugalingam [6].

2 . M a t h e m a t i c a l b a c k g r o u n d

Let(𝑋,𝑑,𝜇)be a metric measure space, where𝜇is a Borel regular measure. Let𝐵(𝑥,𝜌) ⊂ 𝑋be a ball with center𝑥 ∈ 𝑋and radius𝜌 >0.

D e f i n i t i o n 1 ([1, Section 3.1]). A measure𝜇on𝑋is said to be doubling if there exists a constant𝐾, called

the doubling constant, such that0< 𝜇(𝐵(𝑥, 2𝜌)) ≤ 𝐾𝜇(𝐵(𝑥,𝜌)) < +∞for all𝑥 ∈ 𝑋and𝜌 >0. ◀

The following notion of upper gradient has been introduced in order to satisfy the lack of a differentiable structure.

D e f i n i t i o n 2 ([1, Definition 1.13]). A non negative Borel measurable function𝑔is said to be an upper

gradient of function𝑢 ∶ 𝑋 → [−∞,+∞]if, for all compact rectifiable arc lenght parametrized paths𝛾 connecting𝑥and𝑦, we have

(1) |𝑢(𝑥) − 𝑢(𝑦)| ≤ ∫

𝛾

𝑔d𝑠

whenever𝑢(𝑥)and𝑢(𝑦)are both finite and∫_𝛾𝑔d𝑠 = +∞otherwise. ◀

We note that, if𝑔is an upper gradient of function𝑢and𝜙is a non negative Borel measurable function, then𝑔+ 𝜙is still an upper gradient of𝑢. In order to overcome this aspect, we use the following notions that will lead to the definition of the minimal𝑝-weak upper gradient of𝑢.

D e f i n i t i o n 3 ([1, Definition 1.33]). Let𝑝 ∈ [1,+∞[. Let𝛤be a family of paths in𝑋. We say thatinf_𝜙∫_𝑋𝜙^𝑝d𝜇

is the𝑝-modulus of𝛤, where the infimum is taken among all non negative Borel measurable functions𝜙

satisfying∫_𝛾𝜙d𝑠 ≥1, for all rectifiable paths𝛾 ∈ 𝛤. ◀

D e f i n i t i o n 4 ([1, Definition 1.32]). If (1) is satisfied for𝑝-almost all paths𝛾in𝑋, that is, the set of non

constant paths that do not satisfy (1) is of zero𝑝-modulus, then𝑔is said a𝑝-weak upper gradient of𝑢. ◀

The family of weak upper gradients satisfy a result concerning the existence of a minimal element𝑔_ᵆ, that is called the minimal𝑝-weak upper gradient of𝑢.

D e f i n i t i o n 5 ([1, Definition 4.1]). Let𝑝 ∈ [1,+∞[. A metric measure space𝑋supports a(1,𝑝)-Poincaré

inequality if there exist𝐾 >0and𝜆 ≥1such that 1

𝜇(𝐵(𝑥,𝑟))∫

𝐵(𝑥,𝑟)

|𝑢 − 𝑢_𝐵(𝑥_,_𝑟)|d𝜇 ≤ 𝐾𝑟 ( 1 𝜇(𝐵(𝑥,𝜆𝑟))∫

𝐵(𝑥,𝜆𝑟)

𝑔^𝑝ᵆd𝜇)

1 𝑝

for all balls𝐵(𝑥,𝑟) ⊂ 𝑋and for all𝑢 ∈ 𝐿¹_𝑙𝑜𝑐(𝑋), where𝑢𝐵(𝑥,𝑟)= ¹

𝜇(𝐵(𝑥,𝑟))∫_𝐵(𝑥_,_𝑟)𝑢d𝜇. ◀

Let𝑋be a complete metric space equipped with a doubling measure supporting a(1,𝑝)-Poincaré inequality.

We recall the concept of Newtonian space, which is based on the notion of minimal𝑝-weak upper gradient.

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Nastasi

D e f i n i t i o n 6. Let𝑋be a complete metric space equipped with a doubling measure supporting a(1,𝑝)-

Poincaré inequality,𝑝 ∈ [1,+∞]. The Newtonian space𝑁^1,^𝑝(𝑋)is defined by𝑁^1,^𝑝(𝑋) = 𝑉^1,^𝑝(𝑋) ∩ 𝐿^𝑝(𝑋), where𝑉^1,^𝑝(𝑋) = {𝑢 ∶ 𝑢is measurable and𝑔_ᵆ∈ 𝐿^𝑝(𝑋)}.We consider𝑁^1,^𝑝(𝑋)equipped with the norm

‖𝑢‖_𝑁1,𝑝(𝑋)= ‖𝑔_ᵆ‖_𝐿^𝑝_(𝑋)+ ‖𝑢‖_𝐿^𝑝_(𝑋).

We denote with𝑁∗^1,^𝑝(𝑋) = {𝑢 ∈ 𝑁^1,^𝑝(𝑋) ∶ ∫_𝑋𝑢d𝑥 =0}. ◀

The Newtonian space𝑁^1,^𝑝(𝑋)defined above is a complete normed vector space, which generalizes the Sobolev space𝑊^1,^𝑝(𝛺)to a metric setting.

D e f i n i t i o n 7 (see [7]). A Borel set𝐸 ⊂ 𝑋is said to be of finite perimeter if there exists a sequence{𝑢_𝑛}_𝑛∈ℕ

in𝑁^1,1(𝑋)such that𝑢_𝑛 → 𝜒_𝐸in𝐿¹(𝑋)andlim inf_𝑛→+∞∫_𝑋𝑔_ᵆ_𝑛d𝜇 < ∞.The perimeter𝑃_𝐸(𝑋)of𝐸is the infimum of the above limit among all sequences{𝑢_𝑛}as above. For an open set𝑈 ⊂ 𝑋, the perimeter of𝐸 in𝑈is

𝑃𝐸(𝑈) =inf{lim inf

𝑛→+∞∫

𝑋

𝑔ᵆ_𝑛d𝜇 ∶ {𝑢𝑛}𝑛∈ℕ⊂ 𝑁^1,1(𝑈),𝑢𝑛→ 𝜒𝐸∩𝑈in𝐿¹(𝑈)} . ◀

From now on, we consider a bounded domain (non empty, connected open set)𝛺in𝑋with𝑋 ⧵ 𝛺of positive measure such that𝛺is of finite perimeter with perimeter measure𝑃𝛺. Let𝑓 ∶ 𝜕𝛺 → ℝbe a bounded𝑃𝛺-measurable function with∫_𝜕𝛺𝑓d𝑃_𝛺=0. We make the following assumptions on𝛺:

(𝐻₁) There exists a constant𝐾 ≥1such that, for all𝑦 ∈ 𝛺and0< 𝜌 ≤diam(𝛺), we have 𝜇(𝐵(𝑦,𝜌) ∩ 𝛺) ≥ 1

𝐾𝜇(𝐵(𝑦,𝜌)).

(𝐻₂) (Ahlfors codimension 1 regularity of𝑃_𝛺) For all𝑦 ∈ 𝜕𝛺we have that 1

𝐾𝜌𝜇(𝐵(𝑦,𝜌)) ≤ 𝑃_𝛺(𝐵(𝑦,𝜌)) ≤ 𝐾

𝜌𝜇(𝐵(𝑦,𝜌)), where𝐾and𝜌are as in(𝐻₁).

(𝐻₃) (𝛺,𝑑_|𝛺,𝜇_|𝛺)admits a(1,𝑝)-Poincaré inequality with𝜆 =1, where𝑝 ∈]1,+∞[.

D e f i n i t i o n 8 ([4, Definition 4.1]). Let𝛺 ⊂ 𝑋be an open set and let𝑢be a𝜇-measurable function on𝛺. A

function𝑇𝑢 ∶ 𝜕𝛺 →ℝis the trace of𝑢if forℋ-almost every𝑦 ∈ 𝜕𝛺we have

𝜌→lim0⁺

1

𝜇(𝛺 ∩ 𝐵(𝑦,𝜌))∫

𝛺∩𝐵(𝑦,𝜌)

|𝑢 − 𝑇𝑢(𝑦)|d𝜇 =0. ◀

For the existence theorem of the trace operator see Malý and Shanmugalingam [6] and references therein.

Given a Neumann boundary value problem with boundary data𝑓 ≠0and reaction term𝐺, we associate the following functional

𝐽(𝑢) = ∫

𝛺

𝑔ᵆ^𝑝d𝜇 − ∫

𝛺

𝐺(𝑢)d𝜇 + ∫

𝜕𝛺

𝑇𝑢𝑓d𝑃_𝛺 for all𝑢 ∈ 𝑁^1,^𝑝(𝛺).

D e f i n i t i o n 9. A function𝑢₀∈ 𝑁∗^1,^𝑝(𝛺)is a𝑝-harmonic solution to the Neumann boundary value problem

with boundary data𝑓 ≠0and reaction term𝐺if 𝐽(𝑢₀) = ∫

𝛺

𝑔^𝑝ᵆ₀d𝜇 − ∫

𝛺

𝐺(𝑢₀)d𝜇 + ∫

𝜕𝛺

𝑇𝑢₀𝑓d𝑃_𝛺

≤ ∫

𝛺

𝑔^𝑝𝑣d𝜇 − ∫

𝛺

𝐺(𝑣)d𝜇 + ∫

𝜕𝛺

𝑇𝑣𝑓d𝑃_𝛺= 𝐽(𝑣)

for every𝑣 ∈ 𝑁∗^1,^𝑝(𝛺), where𝑔_ᵆ₀,𝑔_𝑣are the minimal𝑝-weak upper gradients of𝑢₀and𝑣in𝛺, respectively, and𝑇𝑢₀and𝑇𝑣are the traces of𝑢₀and𝑣on𝜕𝛺, respectively. ◀

TEMat monogr., 2 (2021) e-ISSN: 2660-6003 89

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Neumann𝑝-Laplacian problems with a reaction term on metric spaces

Later on, in considering the trace𝑇𝑢of𝑢we will omit𝑇and just write𝑢.

Here, we assume that𝐺 ∶ 𝛺 →ℝis defined as𝐺(𝑢) = 𝑐 − |𝑢|^𝛾for all𝑢 ∈ 𝑁^1,^𝑝(𝛺), for some𝑐 > 0and 1< 𝛾 < 𝑝^∗= ^𝑝𝑠

𝑠−𝑝if𝑝 < 𝑠and1< 𝛾 < +∞otherwise.

In the metric setting, we will look for a minimizer of𝐽in the Newtonian space𝑁∗^1,^𝑝(𝛺).

3 . E x i s t e n c e o f a s o l u t i o n a n d a w e a k e r u n i q u e n e s s r e s u l t

The existence of a nontrivial solution to the Neumann boundary value problem with non zero boundary data𝑓and reaction term𝐺is an immediate consequence of the following theorem which shows that𝐽has a minimizer.

T h e o r e m 1 0 . 𝐽has a minimizer in𝑁∗^1,^𝑝(𝛺). If 𝑢₁,𝑢₂ ∈ 𝑁∗^1,^𝑝(𝛺)are two minimizers of 𝐽, then𝑔_ᵆ₁ =𝑔_ᵆ₂

a.e. in𝛺.

4 . B o u n d e d n e s s p r o p e r t y

We show that minimizers are locally bounded near the boundary under appropriate hypothesis on the boundary data𝑓. In order to do so, the following De Giorgi type inequality plays a key role.

L e m m a 1 1. Let 𝑢 ∈ 𝑁∗^1,^𝑝(𝛺)be a minimizer of 𝐽and𝑓 ∈ 𝐿^∞(𝜕𝛺). If 𝑦 ∈ 𝜕𝛺,0< 𝜌 < 𝑅 < ^diam(𝛺)

10 and

𝛼 ∈ℝ, then there is𝐾 ≥1such that the following De Giorgi type inequality

∫

𝛺∩𝐵(𝑦,𝜌)

𝑔_(ᵆ−𝛼)^𝑝

+d𝜇 ≤ 𝐾

(𝑅 − 𝜌)^𝑝∫

𝛺∩𝐵(𝑦,𝑅)

(𝑢 − 𝛼)^𝑝+d𝜇 + 𝐾 ∫

𝜕𝛺∩𝐵(𝑦,𝑅)

|𝑓|(𝑢 − 𝛼)^𝑝+d𝑃𝛺

is satisfied.

T h e o r e m 1 2. Let 0< 𝑅 < ^diam(𝛺)

4 and𝛺𝑅 = {𝑦 ∈ 𝛺 ∶ 𝑑(𝑦,𝜕𝛺) < ^𝑅

2}. If 𝑢 ∈ 𝑁∗^1,^𝑝(𝛺)is a minimizer of 𝐽 and𝑓 ∈ 𝐿^∞(𝜕𝛺), then𝑢 ∈ 𝐿^∞(𝛺𝑅)and 𝑇𝑢 ∈ 𝐿^∞(𝜕𝛺𝑅).

R e f e r e n c e s

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[2] DE GIORGI, Ennio. “Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari”.

In:Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat.3 (1957), pp. 25–43.

[3] KINNUNEN, Juha and SHANMUGALINGAM, Nageswari. “Regularity of quasi-minimizers on metric spaces”. In:Manuscr. Math.105.3 (2001), pp. 401–423.

[4] LAHTI, Panu. “Extensions and traces of functions of bounded variation on metric spaces”. In:J. Math.

Anal. Appl.423.1 (2015), pp. 521–537.

[5] LAHTI, Panu; MALÝ, Lukáš, and SHANMUGALINGAM, Nageswari. “An analog of the Neumann problem for the 1-Laplace equation in the metric setting: existence, boundary regularity, and stability”. In:

Anal. Geom. Metr. Spaces6.1 (2018), pp. 1–31.

[6] MALÝ, Lukáš and SHANMUGALINGAM, Nageswari. “Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity”. In:J. Differ.

Equ.265.6 (2018), pp. 2431–2460.

[7] MIRANDA JR, Michele. “Functions of bounded variation on ‘good’ metric spaces”. In:J. Math. Pures Appl.82.8 (2003), pp. 975–1004.

[8] NASTASI, Antonella. “Weak solution for Neumann (p, q)-Laplacian problem on Riemannian manifold”.

In:J. Math. Anal. Appl.479.1 (2019), pp. 45–61.

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Ricerche di Matematica(2020), pp. 1–16.

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