Characterization of Entire Sequences via Double Orlicz Space

Let Γ denote the space of all entire sequences and ∧ the space of all analytic sequences. This paper is a study of the characterization and general properties of entire sequences via double Orlicz space of ΓM2 of Γ2 establishing some inclusion relations

Critical growth double phase problems

We study Brezis-Nirenberg type Dirichlet problems governed by the double phase operator $-\mathrm{div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)$ and involving a critical nonlinear term of the form $|u|^{p^*-2}u+b(x)|u|^{q^*-2}u$. We prove new compactness and existence results in Musielak-Orlicz Sobolev spaces via variational techniques. The paper is complemented with nonexistence results of Poho\v{z}aev type

The Ideal Convergence of Strongly of Γ

The aim of this paper is to introduce and study a new concept of the Γ2 space via ideal convergence defined by modulus and also some topological properties of the resulting sequence spaces were examined

오리츠 위상 문제의 정칙성

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2022. 8. 변순식.In this thesis, we provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase, which exhibits non-standard growth conditions and non-uniformly elliptic properties. First, we give a unified treatment to show various regularity results for minima of Orlicz multi-phase type functionals with coefficient functions not necessarily Holder continuous even for a lower level of regularity. Moreover, assuming that minima of such functionals belong to better spaces such as Holder or Lebesgue spaces, We address optimal conditions on nonlinearity for each variant under which we build comprehensive regularity results. Second, we prove local Calderon-Zygmund type estimates under the optimal conditions on the nonlinearity for distributional solutions to non-uniformly elliptic equations of Orlicz double phase and multi-phase type in divergence form with the coefficient functions not necessarily Holder continuous. Lastly, we establish an optimal C1,α-regularity for viscosity solutions of a class of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.이 학위논문에서는 오리츠 다상 문제를 포함하고 비표준 성장 조건 및 불균일한 타원형 특성을 나타내는 일반적인 종류의 범함수에 대한 종합적인 정칙성 결과와, 이를 위한 최적의 조건에 대해 조사한다. 우선, 조절 계수가 횔더 연속보다 약화된 경우의 오리츠 다상 범함수의 최소자에 대한 다양한 정칙성 결과를 보이기 위한 통일된 논의를 새롭게 이용한다. 더 나아가, 이러한 범함수의 최소자에 대해 특정 르벡 공간에 포함되거나 횔더 연속이라는 추가 조건이 있을 경우, 정칙성 결과들을 얻기 위해 비선형성에 주어져야 할 최적의 조건들을 찾는다. 두 번째로, 오리츠 이중 위상 및 다중 위상 형태의 발산형 타원 방정식을 고려한다. 비선형성에 최소의 조건을 부여하면서, 이러한 타원 방정식의 분포해에 대한 국소적 칼데론-지그문드 추정을 얻는다. 마지막으로 축퇴/특이 완전 비선형 타원 방정식의 점성 해에 대해 관련 연산자의 최소 정칙성 조건을 찾아, 이 해의 그래디언트 횔더 정칙성을 보인다.Abstract i 1 Introduction 1 2 Preliminaries and auxiliary tools 18 2.1 Notations 18 2.2 Musielak-Orlicz and Musielak-Orlicz-Sobolev spaces 24 2.3 Absence of Lavrentiev phenomenon 27 2.4 Sobolev-Poincare type inequalities 31 2.5 Harmonic type approximation 39 3 Regularity of minima of Orlicz phase functionals 51 3.1 Hypotheses and Main results 51 3.2 Basic regularity results 57 3.2.1 Local boundedness 58 3.2.2 Almost standard Caccioppoli inequality 64 3.2.3 Holder continuity 70 3.2.4 The Harnack inequality 76 3.2.5 Higher integrability results 87 3.3 Comparison estimates 88 3.4 Proof of Theorem 3.1.2 114 3.5 Proof of Theorem 3.1.1 126 3.6 Orlicz double phase problems 131 3.7 Regularity results under additional integrability 159 4 Calderon-Zygmund theory for Orlicz phase problems 169 4.1 Hypotheses and Main results 169 4.2 Homogeneous equations 175 4.2.1 Local boundedness estimates 183 4.2.2 Decay estimates 187 4.2.3 Morrey decay estimate 198 4.2.4 Gradient estimates 202 4.3 Proof of Theorem 4.1.2 209 4.4 Proof of Theorem 4.1.1 223 5 Regularity for degenerate/singular fully nonlinear elliptic equations 225 5.1 Hypotheses and Main results 225 5.2 Basic regularity results 228 5.2.1 Small regime 228 5.2.2 Auxiliary tools 230 5.3 Holder continuity 231 5.4 Approximation 236 5.5 Proof of Theorem 5.1.1 242 Abstract (in Korean) 260 Acknowledgement (in English) 261박

Generalized Orlicz spaces and partial differential equations with application to image restoration

The research area of this thesis is nonlinear functional analysis, a branch of Mathematics which examines questions related to qualitative aspects of solution of a differential equation, such as existence, uniqueness, stability, solvability conditions. Owning to the rapid progress in image processing research involving variational problems, this research work deals with the study of existence and properties of minimizer for image restoration model under Sobolev-Orlicz function space setting. The nature of image restoration that we deal here is noise reduction, where the observed image is assumed to be degraded by a random noise. The noise reduction problem is formulated as a minimization problem consisting of a least squares fit and a regularization term. In the proposed image denoising model, the regularization term represent a double-phase functional that serve the purpose of anisotropic diffusion along with isotropic smoothing, for piecewise smoothing and edge preservation. The mathematical modeling of image restoration problem requires the setting of the domain function space to permit discontinuities of the solution. In this respect, Sobolev-Orlicz function space, which consists of functions having weak derivatives and satisfy certain integrability conditions, provide a favorable framework. For solving such minimization problems, the so-called direct method in the Calculus of Variations is widely used, whose basic topological ingredients are the lower semicontinuity of the functional and the compactness of the lower level sets of the regularization functional. The natural question which then arises here is to study the regularity of such solutions and to establish under which conditions on the data and domain, we have a solution in the sense of distributions. This forms the main objective of my research from the theoretical perspective. Although considerable contributions have been devoted to this challenging question, investigating new approaches under the Sobolev-Orlicz space setting provide new insight into the matter. In this thesis, the study of image restoration problem is carried out in two approaches: variational and PDE-based. The variational approach presents restoration through minimization, where the existence and uniqueness of minimizer is established using the direct Method of the Calculus of Variations. This approach gives information about the qualitative aspects of the model in the Sobolev-Orlicz space setting. In the PDE-based approach, we consider models in the form of heat flow differential equation, where the image is embedded in an evolution process in both space and time dimensions. This yields a quasilinear parabolic boundary value problem. However, due to the degenerate behavior of the PDE, it is not possible to apply general results from classical parabolic equations theory. Thus, to formulate a well-adapted framework, we regularize the PDE using approximations to obtain appropriate solvability conditions. The idea is to construct an approximated boundary problem whose solution converges to the solution of the heat flow problem, under the suitable conditions. Further, to prove the existence and uniqueness of the solution, we derive a few a priori estimates, which gives information about the qualitative behavior of the boundary function. This approach is particularly useful in determining the nature of the domain, where the image corresponds to a feasible solution, that is usually required for numerical purposes. Finally, after proving the existence and uniqueness of the solution, we discretize the problem in order to find a numerical solution. The behaviour and efficiency of the model is then tested and illustrated through numerical experiments. KEYWORDS: image restoration, double-phase, Sobolev-Orlicz space, minimizer, heat flow, PDETämän opinnäytetyön tutkimusalueena on epälineaarinen funktionaalinen analyysi, matematiikan haara, joka tutkii differentiaaliyhtälön ratkaisun laadullisiin näkökohtiin liittyviä kysymyksiä, kuten olemassaoloa, yksikäsitteisyyttä ja stabiilisuutta. Johtuen nopeasta edistymisestä kuvankäsittelytutkimuksessa, joka liittyy variaatio-ongelmiin, tämä tutkimustyö käsittelee kuvan restaurointimallin minimoijan olemassaoloa ja ominaisuuksia Sobolev-Orliczin funktioavaruuksissa. Tässä käsiteltävä kuvan restauroinnin luonne on kohinanvaimennus, jossa oletetaan havaitun kuvan sisältävän satunnaista kohinaa. Kohinanvaimennusongelma on muotoiltu minimoimisongelmaksi, joka koostuu pienimmän neliösumman sovituksesta ja regularisointitermistä. Ehdotetussa kuvan kohinanpoistomallissa regularisointitermi edustaa kaksivaiheista (double phase) funktiota, jossa on sekä anisotrooppinen että isotrooppinen diffuusio paloittaista tasoittamista ja reunan säilyttämistä varten. Kuvan restaurointiongelman matemaattinen mallinnus edellyttää funktioavaruutta, joka mahdollistaa epäjatkuvuudet ratkaisussa. Tässä suhteessa Sobolev-Orlicz-funktioavaruus, joka koostuu funktioista, joilla on heikko derivaatta ja tietyt integroitavuusehdot, tarjoavat suotuisat puitteet. Tällaisten minimointiongelmien ratkaisemiseksi ns. variaatiolaskun suora menetelmä on laajalti käytössä, ja sen topologiset perusainekset ovat funktionaalin alhaalta puolijatkuvuus ja kompaktisuus. Luonnollinen kysymys on tutkia tällaisten ratkaisujen säännöllisyyttä ja pyrkiä määrittämään, millä ehdoilla meillä on ratkaisu distribuution mielessä. Tämä on tutkimukseni päätavoite teoreettisesta näkökulmasta. Vaikka tätä haastavaa kysymystä on tutkittu paljon, uudet lähestymistavat Sobolev-Orliczin avaruuksissa antavat uutta näkemystä asiaan. Tässä opinnäytetyössä kuvan restaurointiongelmaa tutkitaan kahdella lähestymistavalla: variaatio- ja ODY-perustaisesti. Variaatiolaskennassa restaurointia lähestytään minimoinnin kautta, jossa olemassaolo ja minimoinnin yksikäsitteisyys sadaan käyttämällä suoraa variaatiolaskentamenetelmää. Tämä lähestymistapa antaa tietoa mallin laadullisista näkökohdista Sobolev-Orliczin avaruudessa. ODY-pohjaisessa lähestymistavassa tutkimme lämpöyhtälön muotoisia malleja, joissa on evoluutioprosessi sekä tila- että aikaulottuvuuksissa. Tämä tuottaa kvasilineaarisen parabolisen reuna-arvoongelman. ODY:n degeneroidun käyttäytymisen vuoksi siihen ei kuitenkaan voida soveltaa yleisiä tuloksia klassisesta parabolisten yhtälöiden teoriasta. Joten muotoillaksemme sopivan kehyksen, normalisoimme ODY:n käyttämällä approksimaatioita saadaksemme sopivat ratkeavuusehdot. Ideana on rakentaa likimääräiset reuna-arvo ongelmat, joiden ratkaisut suppenevat lämpöyhtälön ratkaisua kohti sopivissa olosuhteissa. Lisäksi todistaaksemme ratkaisun olemassaolon ja yksikäsitteisyyden johdamme a priori arvioita, jotka antavat tietoa rajafunktion laadullisesta käyttäytymisestä. Tämä lähestymistapa on erityisen hyödyllinen määritettäessä alueen luonnetta, jossa kuva vastaa mielekästä ratkaisua, jota yleensä tarvitaan numeerisissa sovelluksissa. Lopuksi, kun olemme todistaneet ratkaisun olemassaolon ja yksikäsitteisyyden, diskretisoimme ongelman löytääksemme numeerisen ratkaisun. Sen jälkeen mallin käyttäytymistä ja tehokkuutta testataan ja havainnollistetaan numeerisilla kokeilla