Maximal regularity estimate for a differential equation with oscillating coefficients

Abstract

xccxvbvpaper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The equation studied is singular as it is defined in an infinite domain, and its coefficients may be unbounded. Its main feature is the rapid growth of the coefficient at of the first derivative of the solution required, therefore the well-developed theory of the Sturm-Liouville equations is not applicable. The equation studied and its multidimensional generalizations arise in the modeling of the Brownian motion of particles, in problems of biology and financial mathematics. Their well-known representatives are the Ornstein-Uhlenbeck and Fokker-Planck-Kolmogorov equations, which have been actively studied since the first half of the twentieth century. On the other hand, projection methods are well known in applications (e.g., Fourier or Laplace transforms), which reduce partial differential equations with coefficients depending on one variable to ordinary differential equations. Therefore, the present study is important for partial derivative equations with unbounded coefficients. In contrast to previous works, the senior and intermediate coefficients of the equation studied can be strongly fluctuating. In the proof of the main theorems, the authors use their earlier result on the correct solvability of the mentioned equation. The paper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The equation studied is singular as it is defined in an infinite domain, and its coefficients may be unbounded. Its main feature is the rapid growth of the coefficient at of the first derivative of the solution required, therefore the well-developed theory of the Sturm-Liouville equations is not applicable. The equation studied and its multidimensional generalizations arise in the modeling of the Brownian motion of particles, in problems of biology and financial mathematics. Their well-known representatives are the Ornstein-Uhlenbeck and Fokker-Planck-Kolmogorov equations, which have been actively studied since the first half of the twentieth century. On the other hand, projection methods are well known in applications (e.g., Fourier or Laplace transforms), which reduce partial differential equations with coefficients depending on one variable to ordinary differential equations. Therefore, the present study is important for partial derivative equations with unbounded coefficients. In contrast to previous works, the senior and intermediate coefficients of the equation studied can be strongly fluctuating. In the proof of the main theorems, the authors use their earlier result on the correct solvability of the mentioned equation. The paper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The equation studied is singular as it is defined in an infinite domain, and its coefficients may be unbounded. Its main feature is the rapid growth of the coefficient at of the first derivative of the solution required, therefore the well-developed theory of the Sturm-Liouville equations is not applicable. The equation studied and its multidimensional generalizations arise in the modeling of the Brownian motion of particles, in problems of biology and financial mathematics. Their well-known representatives are the Ornstein-Uhlenbeck and Fokker-Planck-Kolmogorov equations, which have been actively studied since the first half of the twentieth century. On the other hand, projection methods are well known in applications (e.g., Fourier or Laplace transforms), which reduce partial differential equations with coefficients depending on one variable to ordinary differential equations. Therefore, the present study is important for partial derivative equations with unbounded coefficients. In contrast to previous works, the senior and intermediate coefficients of the equation studied can be strongly fluctuating. In the proof of the main theorems, the authors use their earlier result on the correct solvability of the mentioned equation. The paper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The paper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The equation studied is singular as it is defined in an infinite domain, and its coefficients may be unbounded. Its main feature is the rapid growth of the coefficient at of the first derivative of the solution required, therefore the well-developed theory of the Sturm-Liouville equations is not applicable. The equation studied and its multidimensional generalizations arise in the modeling of the Brownian motion of particles, in problems of biology and financial mathematics. Their well-known representatives are the Ornstein-Uhlenbeck and Fokker-Planck-Kolmogorov equations, which have been actively studied since the first half of the twentieth century. On the other hand, projection methods are well known in applications (e.g., Fourier or Laplace transforms), which reduce partial differential equations with coefficients depending on one variable to ordinary differential equations. Therefore, the present study is important for partial derivative equations with unbounded coefficients. In contrast to previous works, the senior and intermediate coefficients of the equation studied can be strongly fluctuating. In the proof of the main theorems, the authors use their earlier result on the correct solvability of the mentioned equation. The paper considers a second-order differential equation with unbounded coefficients. Sufficient summability conditions with the weight of the solution and its derivatives up to second order are obtained. The equation studied is singular as it is defined in an infinite domain, and its coefficients may be unbounded. Its main feature is the rapid growth of the coefficient at of the first derivative of the solution required, therefore the well-developed theory of the Sturm-Liouville equations is not applicable. The equation studied and its multidimensional generalizations arise in the modeling of the Brownian motion of particles, in problems of biology and financial mathematics. Their well-known representatives are the Ornstein-Uhlenbeck and Fokker-Planck-Kolmogorov equations, which have been actively studied since the first half of the twentieth century. On the other hand, projection methods are well known in applications (e.g., Fourier or Laplace transforms), which reduce partial differential equations with coefficients depending on one variable to ordinary differential equations. Therefore, the present study is important for partial derivative equations with unbounded coefficients. In contrast to previous works, the senior and intermediate coefficients of the equation studied can be strongly fluctuating. In the proof of the main theorems, the authors use their earlier result on the correct solvability of the mentioned equation.