비선형 편미분 방정식에 대한 그래디언트 퍼텐셜 이론

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. 변순식.이 학위 논문에서는 측도데이터를 갖는 비선형 편미분 방정식에 대하여 해의 그라디언트가 각 점 별로 주어진 데이터의 1-리즈 퍼텐셜 가늠을 갖는다는 것을 다양한 비표준 성장조건 하에서 증명하였다. 특히 1-리즈 퍼텐셜이 선형 연산자에 대응하는 퍼텐셜이라는 사실로 인하여 선형화 기법을 통하여 주어진 비선형 연산자를 선형 연산자로 근사하는 방법이 요구된다. 이러한 과정에서 현재까지의 접근법에는 주어진 편미분 방정식의 성장조건이 2차보다 높은 차수인 경우와 낮은 차수인 경우에 대해 근본적인 차이가 있었다. 위와 같은 차이는 이 논문의 4장의 접근법을 통하여 주어진 데이터가 약해의 존재성을 보장할 수 있는 경우에 대하여 극복된다. 그러나 측도데이터를 갖는 방정식에 대해서는 아직까지 위와 같은 통합적인 접근법이 제시되지 않았으며, 이러한 접근법을 제시하는 첫 걸음으로서 5장에서 측도데이터를 갖는 올릭즈 성장조건 방정식에 대하여 칼데론 지그먼드 이론을 설명하고 있다.The objective of this thesis is to provide a sharp gradient potential estimate for nonlinear elliptic problems under non-standard growth assumptions. The estimates have been found from the attempts to develop a unified method for the purpose of obtaining sharp pointwise bounds of the gradient of solutions. First, we obtain gradient potential estimates, by using linearization techniques along with an exit time argument, for two non-autonomous elliptic measure data problems with superquadratic growth. One is variable exponent case the other is mild phase transition case. In gradient potential theory for measure data problems, a unified method is still unknown, that covers both superquadratic and subquadratic cases, because of the difficulty stemming from the absence of energy solutions to such problems. However, once we take energy solutions into account, we devise a new unified method to deal with superquadratic and subquadratic cases simultaneously. In particular, we show partial regularity of the gradient of solutions to subquadratic elliptic systems without the quasi-diagonal structure via Riesz potentials, when the given data belong to suitable Lebesgue spaces to ensure the existence of weak solutions. In the process of a further research on developing a unified method for measure data problems, we establish global Calderon-Zygmund estimates for such problems with general growth via fractional maximal functions.Abstract i 1 Introduction 1 1.1 Measure data problems with polynomial growth . . . . . . . . 2 1.2 Gradient potential theory for non-standard growth problems . 4 1.3 Partial regularity via Riesz potentials . . . . . . . . . . . . . . 6 1.4 Elliptic measure data problems with general growth . . . . . . 8 2 Preliminaries 11 2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Musielak-Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 log-Holder continuity . . . . . . . . . . . . . . . . . . . 16 2.3.2 Monotonicity of vector field A(). . . . . . . . . . . . . 17 2.3.3 Regularity results for limiting equations . . . . . . . . 18 3 Non-autonomous equations 21 3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Comparison estimates . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Basic comparison estimates for (GPT) . . . . . . . . . 27 3.2.2 Basic comparison estimates for (GPX) . . . . . . . . . 36 3.2.3 Higher integrability and further comparison estimates for (GPT) . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Higher integrability and further comparison estimates for (GPX) . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.5 Sequence of comparison estimates for (GPT) . . . . . . 50 3.2.6 Iterative comparison estimates for (GPX) . . . . . . . 60 3.3 Regularity results for homogeneous equation . . . . . . . . . . 68 3.4 Proof of Theorem 3.1.3 . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Gradient continuity via Riesz potentials . . . . . . . . . . . . . 87 4 Subquadratic systems without the quasi-diagonal structure 93 4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2.1 Approximation lemmas. . . . . . . . . . . . . . . . . . 100 4.3 higher integrability . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4 Excess decay estimates . . . . . . . . . . . . . . . . . . . . . . 106 4.4.1 The non-singular case . . . . . . . . . . . . . . . . . . 107 4.4.2 Large measure or oscillatory coefficient . . . . . . . . . 108 4.4.3 Small measure and stable coefficient . . . . . . . . . . . 109 4.4.4 The singular case . . . . . . . . . . . . . . . . . . . . . 115 4.5 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 119 4.5.1 Basic settings . . . . . . . . . . . . . . . . . . . . . . . 119 4.5.2 Iterative lemmas . . . . . . . . . . . . . . . . . . . . . 120 4.5.3 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . 127 5 Measure data problems with general growth 131 5.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Existence of SOLA . . . . . . . . . . . . . . . . . . . . . . . . 134 5.3 Comparison estimates . . . . . . . . . . . . . . . . . . . . . . 140 5.3.1 Technical estimates . . . . . . . . . . . . . . . . . . . . 140 5.3.2 Boundary comparison estimates . . . . . . . . . . . . . 143 5.3.3 Interior comparison estimates . . . . . . . . . . . . . . 154 5.4 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 155 5.5 Calderon-Zygmund theory for integral functionals with p(x)-growth . . . . . . . . . . . . . . 161 5.6 Proof of Theorem 5.5.5 . . . . . . . . . . . . . . . . . . . . . . 164 5.6.1 Auxiliary results for frozen functionals . . . . . . . . . 164 5.6.2 Comparison estimates . . . . . . . . . . . . . . . . . . 168 Bibliography 183 Abstract (in Korean) 193 Acknowledgement (in Korean) 195Docto

Development and Validation of the Korean Rome III Questionnaire for Diagnosis of Functional Gastrointestinal Disorders

BACKGROUND/AIMS: A self-report questionnaire is frequently used to measure symptoms reliably and to distinguish patients with functional gastrointestinal disorders (FGIDs) from those with other conditions. We produced and validated a cross-cultural adaptation of the Rome III questionnaire for diagnosis of FGIDs in Korea. METHODS: The Korean version of the Rome III (Rome III-K) questionnaire was developed through structural translational processes. Subsequently, reliability was measured by a test-retest procedure. Convergent validity was evaluated by comparing self-reported questionnaire data with the subsequent completion of the questionnaire by the physician based on an interview and with the clinical diagnosis. Concurrent validation using the validated Korean version of the Short Form-36 Health Survey (SF-36) was adopted to demonstrate discriminant validity. RESULTS: A total of 306 subjects were studied. Test-retest reliability was good, with a median Cronbach's α value of 0.83 (range, 0.71-0.97). The degree of agreement between patient-administered and physician-administered questionnaires to diagnose FGIDs was excellent; the κ index was 0.949 for irritable bowel syndrome, 0.883 for functional dyspepsia and 0.927 for functional heartburn. The physician's clinical diagnosis of functional dyspepsia showed the most marked discrepancy with that based on the self-administered questionnaire. Almost all SF-36 domains were impaired in participants diagnosed with one of these FGIDs according to the Rome III-K. CONCLUSIONS: We developed the Rome III-K questionnaire though structural translational processes, and it revealed good test-retest reliability and satisfactory construct validity. These results suggest that this instrument will be useful for clinical and research assessments in the Korean population.ope

Chronic atrophic gastritis and intestinal metaplasia surrounding diffuse-type gastric cancer: Are they just bystanders in the process of carcinogenesis?

BACKGROUND: Gastric cancer (GC) is categorized as diffuse- and intestinal-type adenocarcinoma. Intestinal-type GC is associated with chronic gastritis, atrophic gastritis (AG), and intestinal metaplasia (IM), precursors of dysplastic changes. Diffuse-type GC is generally known to undergo de novo carcinogenesis and is not associated with chronic mucosal changes. However, clinically, AG and IM are frequently observed surrounding diffuse-type GC. This study aimed to evaluate the role of AG and IM in diffuse-type GC. METHODS: We retrospectively reviewed the data of patients undergoing surgery for early GC. We divided patients with diffuse-type GC into two groups according to the presence of AG and IM based on Kyoto classification of gastritis. The clinicopathological characteristics were compared between the groups. RESULTS: Among patients with diffuse-type GC, 52.5% patients had AG and 18.4% had severe AG. With regard to IM, 42.1% patients had IM and 17.1% had severe IM. Diffuse-type GC combined with severe AG or IM showed larger tumor size and higher submucosal invasion rate than that without severe AG or IM. However, the lymph node metastasis (LNM) rate was not significantly different between the two groups. In multivariate analysis, severe AG or IM was not an independent risk factor for LNM. CONCLUSIONS: Severe AG or IM surrounding diffuse-type gastric cancer suggests a collapse of normal mucosal barriers and leads to the spread of cancer cells. Although the association between chronic mucosal changes and LNM is unclear, more caution is needed during endoscopy especially for complete resection of diffuse-type GC with these features.ope

Multicenter, Randomized, Placebo-controlled Trial to Evaluate the Efficacy and Safety of a Controlled-release, Once-daily UIC201609/UIC201610 Combination Therapy for Functional Dyspepsia: Preliminary Study

Backgrounds/aims: Functional dyspepsia is a disease involving a range of upper gastrointestinal symptoms derived from various pathophysiologies. Tablets containing a combination of rabeprazole and controlled-release (CR) mosapride were recently developed. To investigate a more effective treatment, this trial evaluated the efficacy and safety of UIC201609/UIC201610 as a preliminary study. Methods: A multicenter, double-blind, randomized study was performed on 30 subjects. UIC201609/UIC201610 (combination of rabeprazole and CR mosapride) was the case group, and the two control groups were rabeprazole 10 mg once a day and mosapride 15 mg CR tablet once a day. As a primary efficacy endpoint of the study, the changes in the total score of eight items of the Nepean Dyspepsia Index-Korean version were analyzed at 2 weeks and 4 weeks. The outcomes regarding safety were collected. Results: The total symptom score of Nepean Dyspepsia Index-Korean decreased in the rabeprazole single group (29.4±17.1), mosapride CR single group (33.4±15.6), and UIC201609/UIC201610 group (33.4±11.8) at 4 weeks without significant differences. On the other hand, the UIC201609/UIC201610 combination group showed more score reduction of pain in the upper abdomen, burning in the upper abdomen compared to each control group, but it did not reach statistical significance. No difference was found in safety analysis. Conclusions: UIC201609/UIC201610 once daily showed some improvement in epigastric pain and dyspepsia in patients with functional dyspepsia, but there was no significance. Further study based on the advanced clinical trial design will be needed to confirm the efficacy of UIC201609/UIC201610 combination therapy in the future.ope

Reversibility of Endoscopic Features after Treatment for Eosinophilic Esophagitis

Purpose: The prevalence and incidence of eosinophilic esophagitis (EoE) are increasing worldwide. Despite increased understanding of inflammatory pathogenesis, changes in endoscopic features after treatment of EoE have not been clearly described. We aimed to investigate the reversibility of endoscopic features of EoE after treatment. Materials and methods: Out of 58 adult subjects who were diagnosed with EoE at the Yonsei University Health System from July 2006 to August 2019, we recruited 33 subjects (30 males; mean age: 42 years) whose pre-treatment and post-treatment endoscopic images were available. Endoscopic features included both inflammatory and fibrostenotic features. Exudate, edema, furrow, and crepe paper-like mucosa were classified as inflammatory features. Ring and stricture were classified as fibrostenotic features. We compared changes in endoscopic features after treatment for EoE. Results: After treatment, clinical symptoms improved in all patients. The following endoscopic features were observed before treatment: furrow (81.8%), edema (90.9%), exudate (42.4%), ring (27.3%), crepe paper-like mucosa (15.2%), and stricture (3.0%). Endoscopic remission was achieved in 21 patients (63.6%). Inflammatory features were reversible (72.7%, p<0.001), whereas fibrostenotic features were not (10%, p=0.160). Exudate had resolved in 92.9% of patients, edema in 70% and furrow in 88.9%. Ring and stricture persisted in almost all of the patients (9/10) who had these endoscopic features before treatment. Conclusion: We outlined the reversibility of endoscopic inflammatory features of EoE. Fibrostenotic features were irreversible after esophageal remodeling in patients with EoE. However, further validation studies with long-term follow-up are needed.ope

Non-Ampullary Duodenal Tumors

Non-ampullary duodenal tumors refer to duodenal lesions that originate at sites other than the ampulla of Vater. They differ from ampullary duodenal tumors, which are typically associated with obstructive symptoms. Non-ampullary duodenal tumors are rare; therefore, standardized clinical management, including diagnostic and therapeutic modalities and the follow-up strategy remain unclear. An increase in the numbers of patients undergoing endoscopy during medical checkups and technological advances in endoscopic procedures, including high-resolution and image-enhanced endoscopy have led to an increase in detection rates of duodenal tumors recently. Currently, endoscopic resection of duodenal tumors is widely used and is a preferred therapeutic alternative to surgical excision. Some duodenal tumors of high malignant potential are associated with poor prognosis; therefore, accurate diagnosis and timely management are important. In this article, we review the current knowledge regarding non-ampullary duodenal tumors and the therapeutic approaches to these lesions.ope

Peroral Endoscopic Myotomy, a Well-Established, Efficacious, and Safe Treatment Option for Achalasia: Is the History of Previous Treatment a Hurdle or Not?

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Peri-operative Inflammatory Marker as a Predictive Factor for Prolonged Post-operative Ileus After Gastrectomy for Gastric Cancer

Background/aims: Although prolonged post-operative ileus (PPOI) is an important factor for the prolonged length of post-operative hospital stay, there is still a lack of effective predictive and therapeutic methods for PPOI. Previous studies reported that increased inflammatory markers, such as C-reactive protein (CRP) level and neutrophil to lymphocyte ratio (NLR), are associated with malignancies. The aim of our study is to elucidate the association between peri-operative inflammatory markers and PPOI after gastrectomy for gastric cancer. Methods: We enrolled patients who received gastrectomy for gastric cancer from June 2013 to January 2016 at a single tertiary referral center in Seoul, Korea. We evaluated peri-operative inflammatory markers, including CRP level, NLR, and platelet to lymphocyte ratio (PLR) of enrolled patients. We compared these data between control group and PPOI group. Results: A total of 390 subjects were enrolled in this study, and 132 patients (33.8%) showed PPOI. In univariate analysis, preoperative CRP level and NLR, post-operative day (POD) 1 CRP level, NLR, and PLR, and POD3 CRP level, NLR, and PLR were significantly associated with PPOI. In multivariate analysis, preoperative NLR (P = 0.014), POD1 NLR (P = 0.019), POD3 CRP (P = 0.004), and POD3 NLR (P = 0.008) were independent risk factors for PPOI. Conclusions: Peri-operative inflammatory markers, such as CRP level and NLR, are useful predictive factors for PPOI who received gastrectomy for gastric cancer. Moreover, prophylactic antibiotics and anti-inflammatory drugs can be preventive and therapeutic agents for PPOI.ope

Randomised phase 3 trial: tegoprazan, a novel potassium-competitive acid blocker, vs. esomeprazole in patients with erosive oesophagitis

BACKGROUND: Tegoprazan is a novel potassium-competitive acid blocker that has a fast onset of action and can control gastric pH for a prolonged period, which could offer clinical benefit in acid-related disorders. AIM: To confirm the non-inferiority of tegoprazan to esomeprazole in patients with erosive oesophagitis (EE). METHODS: In this multicentre, randomised, double-blind, parallel-group comparison study, 302 Korean patients with endoscopically confirmed EE (Los Angeles Classification Grades A-D) were randomly allocated to either tegoprazan (50 or 100 mg) or esomeprazole (40 mg) treatment groups for 4 or 8 weeks. The primary endpoint was the cumulative proportion of patients with healed EE confirmed by endoscopy up to 8 weeks from treatment initiation. Symptoms, safety and tolerability were also assessed. RESULTS: The cumulative healing rates at week 8 were 98.9% (91/92), 98.9% (90/91) and 98.9% (87/88) for tegoprazan 50 mg, tegoprazan 100 mg and esomeprazole 40 mg, respectively. Both doses of tegoprazan were non-inferior to esomeprazole 40 mg. The incidence of adverse events was comparable among the groups, and tegoprazan was well-tolerated. CONCLUSION: Once daily administration of tegoprazan 50 or 100 mg showed non-inferior efficacy in healing EE and tolerability to that of esomeprazole 40 mg.ope

Dysphagia in Gastrointestinal Cancer Patients

Dysphagia is one of the common symptoms that are encountered in clinical practice. However, dysphagia is still crucial and must be thoroughly investigated because it may be a key symptom of several malignancies. There are two types of dysphagia, oropharyngeal and esophageal dysphagia. Esophageal dysphagia can be caused by esophageal neuromuscular motility disorder, various inflammatory disorders, and also extrinsic or intrinsic structural lesions such as esophageal cancer. This article focuses on malignant esophageal dysphagia, including its causes, risk factors, clinical symptoms, and management.ope